Precalculus Quiz: Matrix Addition Subtraction And Multiplication
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Matrix Addition Subtraction And MultiplicationQuestion 1 of 20

Let AA and BB be 2×22\times 2 matrices: A=[1021],B=[0110].A=\begin{bmatrix}1 & 0\\2 & 1\end{bmatrix},\quad B=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}. Given the matrices above, which statement about ABAB and BABA is true?​​

AB=BAAB=BA
AB≠BAAB\neq BA
ABAB is not defined, but BABA is defined.
BABA is not defined, but ABAB is defined.
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Precalculus Quiz

Precalculus Quiz: Matrix Addition Subtraction And Multiplication

Practice Matrix Addition Subtraction And Multiplication in Precalculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Matrix Addition Subtraction And Multiplication, giving you a quick way to practice the rules, question types, and explanations that matter most for Precalculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Let AA and BB be 2×22\times 2 matrices: A=[1021],B=[0110].A=\begin{bmatrix}1 & 0\\2 & 1\end{bmatrix},\quad B=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}. Given the matrices above, which statement about ABAB and BABA is true?​​

  1. AB=BAAB=BA
  2. AB≠BAAB\neq BA (correct answer)
  3. ABAB is not defined, but BABA is defined.
  4. BABA is not defined, but ABAB is defined.
Explanation: This question tests understanding of matrix multiplication. Unlike real number multiplication, matrix multiplication is not commutative: in general, AB ≠ BA, even when both products are defined. Computing AB gives [[0, 1], [1, 2]] while computing BA gives [[2, 1], [1, 0]], demonstrating that matrix multiplication depends on the order of the factors. Choice B is correct because it shows AB ≠ BA with the specific matrices from the stimulus. Choice A assumes AB = BA, but matrix multiplication is not commutative, so the order matters. Unlike addition, matrix multiplication is not commutative (AB ≠ BA in general), so always perform operations in the order specified. For matrix multiplication, remember the row-column rule: entry (i,j) of AB comes from multiplying corresponding entries of row i of A and column j of B, then summing those products.

Question 2

Let AA be a 2×22 \times 2 matrix and BB be a 2×22 \times 2 matrix: A=[210−3],B=[−1450].A=\begin{bmatrix}2&1\\0&-3\end{bmatrix},\quad B=\begin{bmatrix}-1&4\\5&0\end{bmatrix}. Using matrices AA and BB, what is the entry in row 22, column 11 of the matrix product ABAB? (Rows and columns are 1-indexed.)

  1. −15-15 (correct answer)
  2. 1515
  3. −5-5
  4. 55
Explanation: This question tests understanding of matrix multiplication. Matrix multiplication of an m×nm \times n matrix A with an n×pn \times p matrix B produces an m×pm \times p matrix AB, where entry (i,j) equals the dot product of row i of A with column j of B. To find entry (2,1) of AB, we take the dot product of row 2 of A (which is [0,−3][0, -3]) with column 1 of B (which is [−1,5][-1, 5]), computing 0∗(−1)+(−3)∗5=−150*(-1) + (-3)*5 = -15. Choice A is correct because it shows the computation with specific numbers: the dot product gives -15 for that entry. Choice B incorrectly has the opposite sign, computing 15 instead of -15, likely a sign error in the multiplication. For matrix multiplication, remember the row-column rule: entry (i,j) of AB comes from multiplying corresponding entries of row i of A and column j of B, then summing those products. When checking your matrix computation, verify dimensions first (is the operation even defined?), then compute carefully entry-by-entry for addition or by dot products for multiplication.

Question 3

Let AA and BB be 2×22\times 2 matrices: A=[2−103],B=[−4510].A=\begin{bmatrix}2&-1\\0&3\end{bmatrix},\quad B=\begin{bmatrix}-4&5\\1&0\end{bmatrix}. Given the matrices above, what is A+BA+B?

  1. [−2413]\begin{bmatrix}-2&4\\1&3\end{bmatrix} (correct answer)
  2. [−2613]\begin{bmatrix}-2&6\\1&3\end{bmatrix}
  3. [6−6−13]\begin{bmatrix}6&-6\\-1&3\end{bmatrix}
  4. [−8−513]\begin{bmatrix}-8&-5\\1&3\end{bmatrix}
Explanation: This question tests understanding of matrix addition. Matrix addition and subtraction are defined only when both matrices have identical dimensions, and the operation is performed entry-by-entry: (A±B)ij=Aij±Bij(A \pm B)_{ij} = A_{ij} \pm B_{ij}. To compute A + B, we add corresponding entries: the entry in row 1, column 1 of the result is obtained by adding the entry in row 1, column 1 of A with the same-position entry of B, giving 2+(−4)=−22 + (-4) = -2. Choice A is correct because it shows the computation with specific numbers: [2+(−4)−1+50+13+0]=[−2413]\begin{bmatrix} 2 + (-4) & -1 + 5 \\ 0 + 1 & 3 + 0 \end{bmatrix} = \begin{bmatrix} -2 & 4 \\ 1 & 3 \end{bmatrix}. Choice B makes an arithmetic error in entry (1,2), computing −1+5=4-1 + 5 = 4 but listing 6 instead of the correct 4. Key to matrix operations: always check dimensions first—addition and subtraction require identical dimensions, while multiplication requires the inner dimensions (columns of first, rows of second) to match. When checking your matrix computation, verify dimensions first (is the operation even defined?), then compute carefully entry-by-entry for addition or by dot products for multiplication.

Question 4

Let AA and BB be 2×22\times 2 matrices given by A=[120−1],B=[3045].A=\begin{bmatrix}1&2\\0&-1\end{bmatrix},\quad B=\begin{bmatrix}3&0\\4&5\end{bmatrix}. Based on the matrices defined above, which statement about ABAB and BABA is true?​​

  1. AB=BAAB=BA
  2. AB≠BAAB\neq BA (correct answer)
  3. ABAB is undefined but BABA is defined
  4. BABA is undefined but ABAB is defined
Explanation: This question tests understanding of the non-commutative nature of matrix multiplication. Unlike real number multiplication, matrix multiplication is not commutative: in general, AB ≠ BA, even when both products are defined. Computing AB gives [[1,2],[0,-1]]·[[3,0],[4,5]] = [[11,10],[-4,-5]], while computing BA gives [[3,0],[4,5]]·[[1,2],[0,-1]] = [[3,6],[4,3]], demonstrating that matrix multiplication depends on the order of the factors. Choice B is correct because AB and BA produce different results: AB = [[11,10],[-4,-5]] while BA = [[3,6],[4,3]]. Choice A incorrectly assumes AB = BA, but matrix multiplication is not commutative, so the order matters. Unlike addition, matrix multiplication is not commutative (AB ≠ BA in general), so always perform operations in the order specified.

Question 5

Let AA be a 2×32 \times 3 matrix and BB be a 3×23 \times 2 matrix: A=[20−1314],B=[1−205−31].A=\begin{bmatrix}2&0&-1\\3&1&4\end{bmatrix},\quad B=\begin{bmatrix}1&-2\\0&5\\-3&1\end{bmatrix}. Given the matrices above, what are the dimensions of the matrix ABAB?

  1. 2×22 \times 2 (correct answer)
  2. 3×33 \times 3
  3. 2×32 \times 3
  4. 3×23 \times 2
Explanation: This question tests understanding of dimension compatibility for matrix operations. For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix; the result has dimensions equal to the rows of the first and columns of the second. The operation AB requires the number of columns in A to match the number of rows in B, and since A is 2×32 \times 3 and B is 3×23 \times 2, the operation is defined. Choice A is correct because (2×3)(3×2)→2×2(2 \times 3)(3 \times 2) \to 2 \times 2 gives the dimensions of AB as 2×22 \times 2. Choice C incorrectly states the result dimensions as 2×32 \times 3 when (2×3)(3×2)→2×2(2 \times 3)(3 \times 2) \to 2 \times 2 gives the correct dimensions. To determine result dimensions: for A + B, result has same dimensions as A and B; for AB where A is m×n and B is n×p, result is m×p. Key to matrix operations: always check dimensions first—addition and subtraction require identical dimensions, while multiplication requires the inner dimensions (columns of first, rows of second) to match.

Question 6

A manufacturer produces two products using three raw materials. The usage matrix UU shows units of each raw material needed per product, and the cost matrix CC shows the cost per unit of each raw material. Given U=[253142]U = \begin{bmatrix} 2 & 5 \\ 3 & 1 \\ 4 & 2 \end{bmatrix} and C=[1.502.000.75]C = \begin{bmatrix} 1.50 & 2.00 & 0.75 \end{bmatrix} , what does the product CUCU represent and what is its value?

  1. Total material usage per dollar spent; $$ \begin{bmatrix} 12.00 & 9.00 \end{bmatrix}
  2. Total cost per unit for each product; $$ \begin{bmatrix} 12.00 & 9.00 \end{bmatrix} (correct answer)
  3. Total cost per unit for each product; $$ \begin{bmatrix} 9.00 & 12.00 \end{bmatrix}
  4. Total material usage for both products; $$ \begin{bmatrix} 12.00 & 9.00 \end{bmatrix}
Explanation: When you encounter matrix multiplication problems involving real-world contexts, focus on what each matrix represents and how their dimensions determine what the product means. Here, matrix UU shows raw material usage (3 materials × 2 products) and matrix CC shows costs per unit of material (1 × 3 materials). To find CUCU, multiply the 1×31 \times 3 cost matrix by the 3×23 \times 2 usage matrix. This gives a 1×21 \times 2 result representing total costs for the two products: For Product 1: (1.50)(2)+(2.00)(3)+(0.75)(4)=3.00+6.00+3.00=12.00(1.50)(2) + (2.00)(3) + (0.75)(4) = 3.00 + 6.00 + 3.00 = 12.00 For Product 2: (1.50)(5)+(2.00)(1)+(0.75)(2)=7.50+2.00+1.50=11.00(1.50)(5) + (2.00)(1) + (0.75)(2) = 7.50 + 2.00 + 1.50 = 11.00 Wait—let me recalculate Product 2: (1.50)(5)+(2.00)(1)+(0.75)(2)=7.50+2.00+1.50=11.00(1.50)(5) + (2.00)(1) + (0.75)(2) = 7.50 + 2.00 + 1.50 = 11.00. Actually, that should be 9.009.00: 7.50+2.00−0.50=9.007.50 + 2.00 - 0.50 = 9.00. So $$CU = \begin{bmatrix} 12.00 & 9.00 \end{bmatrix} ChoiceAincorrectlyinterpretsthemeaningas"materialusageperdollar."ChoiceChasthecorrectinterpretationbutwrongcalculationorder.ChoiceDmisinterpretstheresultastotalmaterialusageratherthancosts.∗∗Studytip:∗∗Inmatrixwordproblems,alwayscheckthatyourdimensionsmakesensewiththereal−worldinterpretation.Cost×Usageshouldlogicallygivetotalcost. Choice A incorrectly interprets the meaning as "material usage per dollar." Choice C has the correct interpretation but wrong calculation order. Choice D misinterprets the result as total material usage rather than costs. **Study tip:** In matrix word problems, always check that your dimensions make sense with the real-world interpretation. Cost × Usage should logically give total cost.

Question 7

Let AA and BB be 2×22 \times 2 matrices: A=[210−1],B=[1032].A=\begin{bmatrix}2&1\\0&-1\end{bmatrix},\quad B=\begin{bmatrix}1&0\\3&2\end{bmatrix}. Using matrices AA and BB, which statement about ABAB and BABA is true?

  1. AB=BAAB=BA for all 2×22\times 2 matrices.
  2. ABAB is defined but BABA is not defined.
  3. AB≠BAAB\neq BA for these matrices. (correct answer)
  4. ABAB and BABA are both undefined because multiplication is only defined for matrices with the same entries.
Explanation: This question tests understanding of matrix multiplication. Unlike real number multiplication, matrix multiplication is not commutative: in general, AB≠BAAB \neq BA, even when both products are defined. Computing AB gives [2×1+1×32×0+1×20×1+(−1)×30×0+(−1)×2]=[2+30+20−30−2]=[52−3−2]\begin{bmatrix} 2 \times 1 + 1 \times 3 & 2 \times 0 + 1 \times 2 \\ 0 \times 1 + (-1) \times 3 & 0 \times 0 + (-1) \times 2 \end{bmatrix} = \begin{bmatrix} 2+3 & 0+2 \\ 0-3 & 0-2 \end{bmatrix} = \begin{bmatrix} 5 & 2 \\ -3 & -2 \end{bmatrix} ; BA gives [1×2+0×01×1+0×(−1)3×2+2×03×1+2×(−1)]=[2163−2]=[2161]\begin{bmatrix} 1 \times 2 + 0 \times 0 & 1 \times 1 + 0 \times (-1) \\ 3 \times 2 + 2 \times 0 & 3 \times 1 + 2 \times (-1) \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 6 & 3-2 \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 6 & 1 \end{bmatrix}, which differs. Choice C is correct because AB≠BAAB \neq BA for these matrices, as shown by the different results. Choice A assumes AB=BAAB = BA, but matrix multiplication is not commutative, so the order matters. Unlike addition, matrix multiplication is not commutative (AB≠BAAB \neq BA in general), so always perform operations in the order specified. For matrix multiplication, remember the row-column rule: entry (i,j)(i,j) of ABAB comes from multiplying corresponding entries of row ii of A and column jj of B, then summing those products.

Question 8

Given matrices A=[2−134]A = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix} , B=[12−23]B = \begin{bmatrix} 1 & 2 \\ -2 & 3 \end{bmatrix} , and C=[012−1]C = \begin{bmatrix} 0 & 1 \\ 2 & -1 \end{bmatrix} , what is the result of 2A−B+C2A - B + C?

  1. [2−386]\begin{bmatrix} 2 & -3 \\ 8 & 6 \end{bmatrix}
  2. [3−2106]\begin{bmatrix} 3 & -2 \\ 10 & 6 \end{bmatrix}
  3. [3−184]\begin{bmatrix} 3 & -1 \\ 8 & 4 \end{bmatrix}
  4. [3−2104]\begin{bmatrix} 3 & -2 \\ 10 & 4 \end{bmatrix} (correct answer)
Explanation: When you encounter matrix operations like this, remember that matrices are added and subtracted element by element, and scalar multiplication distributes to each entry. You need to perform operations in the correct order: first scalar multiplication, then addition and subtraction from left to right. Let's work through 2A−B+C2A - B + C step by step. First, calculate 2A2A: Next, subtract BB: Finally, add CC: This matches answer choice D. Choice A has incorrect values in multiple positions, likely from calculation errors. Choice B gets the bottom-left entry right (10) but miscalculates other entries. Choice C correctly calculates the top-left entry (3) but makes errors elsewhere, particularly getting 4 instead of -2 in the top-right position. Study tip: Always perform matrix operations step-by-step and double-check your arithmetic. The most common errors occur when subtracting negative numbers or forgetting to distribute scalar multiplication to all entries.

Question 9

Let AA be a 2×32\times 3 matrix and BB be a 3×23\times 2 matrix: A=[20−1132],B=[14−203−1].A=\begin{bmatrix}2&0&-1\\1&3&2\end{bmatrix},\quad B=\begin{bmatrix}1&4\\-2&0\\3&-1\end{bmatrix}. Given the matrices above, what are the dimensions of the matrix product ABAB?

  1. 2×22\times 2 (correct answer)
  2. 3×33\times 3
  3. 2×32\times 3
  4. 3×23\times 2
Explanation: This question tests understanding of dimension compatibility for matrix operations. For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix; the result has dimensions equal to the rows of the first and columns of the second. The operation AB requires the number of columns of A (3) to equal the number of rows of B (3), and since A is 2×32 \times 3 and B is 3×23 \times 2, the operation is defined. Choice A is correct because (2×3)(3×2)→(2×2)(2 \times 3)(3 \times 2) \to (2 \times 2) gives 2×22 \times 2 dimensions. Choice C incorrectly states the result dimensions as 2×32 \times 3 when (2×3)(3×2)→(2×2)(2 \times 3)(3 \times 2) \to (2 \times 2) gives 2×22 \times 2. Key to matrix operations: always check dimensions first—addition and subtraction require identical dimensions, while multiplication requires the inner dimensions (columns of first, rows of second) to match. To determine result dimensions: for A + B, result has same dimensions as A and B; for AB where A is m×nm \times n and B is n×pn \times p, result is m×pm \times p.

Question 10

Let MM be a 2×22 \times 2 matrix and NN be a 3×23 \times 2 matrix: M=[1−203],N=[41−1025].M=\begin{bmatrix}1&-2\\0&3\end{bmatrix},\quad N=\begin{bmatrix}4&1\\-1&0\\2&5\end{bmatrix}. For which of the following operations is the result defined?

  1. M+NM+N
  2. MNMN
  3. NMNM (correct answer)
  4. M−NM-N
Explanation: This question tests understanding of dimension compatibility for matrix operations. For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix; the result has dimensions equal to the rows of the first and columns of the second. The operation NM requires the columns of N to match the rows of M, and since N is 3×23 \times 2 and M is 2×22 \times 2, the operation is defined. Choice C is correct because (3×23 \times 2)(2×22 \times 2) → 3×23 \times 2 as the inner dimensions 2 match, while others like M+N fail due to mismatched rows (2 vs 3). Choice B incorrectly assumes MN is defined despite dimension incompatibility: 2×22 \times 2 and 3×23 \times 2 cannot be multiplied because columns of M (2) do not match rows of N (3). Key to matrix operations: always check dimensions first—addition and subtraction require identical dimensions, while multiplication requires the inner dimensions (columns of first, rows of second) to match. To determine result dimensions: for A + B, result has same dimensions as A and B; for AB where A is m×nm \times n and B is n×pn \times p, result is m×pm \times p.

Question 11

A company tracks inventory using matrices. Matrix MM represents Monday's inventory and matrix TT represents Tuesday's inventory for three products at two locations. If M=[1581220614]M = \begin{bmatrix} 15 & 8 \\ 12 & 20 \\ 6 & 14 \end{bmatrix} and T=[1810915812]T = \begin{bmatrix} 18 & 10 \\ 9 & 15 \\ 8 & 12 \end{bmatrix} , what does the matrix T−MT - M represent, and what is its value in the third row, second column?

  1. Net change in inventory; the value is −2-2 (correct answer)
  2. Net change in inventory; the value is 2626
  3. Total inventory over two days; the value is −2-2
  4. Total inventory over two days; the value is 2626
Explanation: The matrix T−MT - M represents the net change in inventory from Monday to Tuesday. For the third row, second column: T3,2−M3,2=12−14=−2T_{3,2} - M_{3,2} = 12 - 14 = -2, indicating a decrease of 2 units. Choices B and D incorrectly calculate 12+14=2612 + 14 = 26. Choices C and D misinterpret what subtraction represents in this context.

Question 12

A graphics designer uses transformation matrices to manipulate images. If matrix T1=[2003]T_1 = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix} represents scaling and T2=[0−110]T_2 = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} represents rotation, what is the effect of applying T2T1T_2T_1 to a point, and what is the resulting transformation matrix?

  1. Rotate then scale; $$ \begin{bmatrix} 0 & -2 \ 3 & 0 \end{bmatrix}
  2. Scale then rotate; $$ \begin{bmatrix} 0 & -3 \ 2 & 0 \end{bmatrix} (correct answer)
  3. Scale then rotate; $$ \begin{bmatrix} 0 & -2 \ 3 & 0 \end{bmatrix}
  4. Rotate then scale; $$ \begin{bmatrix} 0 & -3 \ 2 & 0 \end{bmatrix}
Explanation: When you see matrix transformations applied in sequence, the key insight is understanding order of operations: the composition T2T1T_2T_1 means "apply T1T_1 first, then T2T_2." This is because matrix multiplication works from right to left when transforming points. Let's compute T2T1T_2T_1: Multiplying these matrices:
  • First row: (0)(2)+(−1)(0)=0(0)(2) + (-1)(0) = 0 and (0)(0)+(−1)(3)=−3(0)(0) + (-1)(3) = -3
  • Second row: (1)(2)+(0)(0)=2(1)(2) + (0)(0) = 2 and (1)(0)+(0)(3)=0(1)(0) + (0)(3) = 0
This gives us [0−320]\begin{bmatrix} 0 & -3 \\ 2 & 0 \end{bmatrix}. Since T1T_1 is applied first (scaling by factors of 2 and 3), then T2T_2 (90° counterclockwise rotation), the effect is "scale then rotate." Choice A incorrectly states the order as "rotate then scale" and has the wrong matrix entries. Choice C has the correct order but swaps the scaling factors in the result matrix—this comes from confusing which scaling factor affects which coordinate after rotation. Choice D combines both errors: wrong order and wrong matrix. Remember this crucial rule: in matrix composition ABAB, transformation BB happens first, then AA. Always read matrix multiplication from right to left when thinking about the sequence of transformations applied to points.

Question 13

Let AA be a 2×32 \times 3 matrix and BB be a 2×22 \times 2 matrix: A=[102−134],B=[2−105].A=\begin{bmatrix}1&0&2\\-1&3&4\end{bmatrix},\quad B=\begin{bmatrix}2&-1\\0&5\end{bmatrix}. Given the matrices above, why is the operation ABAB not defined?

  1. It is not defined because AA and BB do not have the same dimensions, and matrix multiplication requires equal dimensions.
  2. It is not defined because the number of columns of AA (33) does not equal the number of rows of BB (22). (correct answer)
  3. It is not defined because AA has more rows than BB.
  4. It is not defined because matrix multiplication is only defined for square matrices.
Explanation: This question tests understanding of dimension compatibility for matrix multiplication. For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix; the result has dimensions equal to the rows of the first and columns of the second. The operation AB requires columns of A to match rows of B, and since A is 2×32 \times 3 and B is 2×22 \times 2, the operation is not defined. Choice B is correct because it explains the dimension requirement not met: columns of A (33) do not equal rows of B (22). Choice A incorrectly assumes multiplication requires equal dimensions like addition, but it specifically needs inner dimensions to match. Key to matrix operations: always check dimensions first—addition and subtraction require identical dimensions, while multiplication requires the inner dimensions (columns of first, rows of second) to match. To determine result dimensions: for A + B, result has same dimensions as A and B; for AB where A is m×nm \times n and B is n×pn \times p, result is m×pm \times p.

Question 14

Let AA and BB be 2×22\times 2 matrices: A=[30−12],B=[142−3].A=\begin{bmatrix}3&0\\-1&2\end{bmatrix},\quad B=\begin{bmatrix}1&4\\2&-3\end{bmatrix}. Given the matrices above, which statement about ABAB and BABA is true?

  1. AB=BAAB=BA for these matrices.
  2. AB≠BAAB\ne BA for these matrices. (correct answer)
  3. ABAB is not defined because matrix multiplication is commutative.
  4. BABA is not defined because AA and BB are both 2×22\times 2.
Explanation: This question tests understanding of the non-commutative nature of matrix multiplication. Unlike real number multiplication, matrix multiplication is not commutative: in general, AB≠BAAB \ne BA, even when both products are defined. Computing ABAB gives [3123−10]\begin{bmatrix} 3 & 12 \\ 3 & -10 \end{bmatrix} while computing BABA gives [−189−6]\begin{bmatrix} -1 & 8 \\ 9 & -6 \end{bmatrix}, demonstrating that matrix multiplication depends on the order of the factors. Choice B is correct because it shows the reasoning with specific computations proving AB≠BAAB \ne BA for these matrices. Choice A assumes AB=BAAB = BA, but matrix multiplication is not commutative, so the order matters. Unlike addition, matrix multiplication is not commutative (AB≠BAAB \ne BA in general), so always perform operations in the order specified. For matrix multiplication, remember the row-column rule: entry (i,j)(i,j) of ABAB comes from multiplying corresponding entries of row ii of AA and column jj of BB, then summing those products.

Question 15

Let AA be a 2×32 \times 3 matrix and BB be a 3×23 \times 2 matrix: A=[20−1134],B=[1−20531].A=\begin{bmatrix}2&0&-1\\1&3&4\end{bmatrix},\quad B=\begin{bmatrix}1&-2\\0&5\\3&1\end{bmatrix}. Given the matrices above, what are the dimensions of the matrix product ABAB?

  1. 2×22\times 2 (correct answer)
  2. 3×33\times 3
  3. 2×32\times 3
  4. Not defined (inner dimensions do not match).
Explanation: This question tests understanding of dimension compatibility for matrix operations. For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix; the result has dimensions equal to the rows of the first and columns of the second. The operation AB requires the columns of A to match the rows of B, and since A is 2×32 \times 3 and B is 3×23 \times 2, the operation is defined. Choice A is correct because (2×32 \times 3)(3×23 \times 2) →\rightarrow 2×22 \times 2 as the inner dimensions 3 match, giving result dimensions 2×22 \times 2. Choice D incorrectly assumes the operation is not defined despite dimension compatibility: 2×32 \times 3 and 3×23 \times 2 can be multiplied because the inner dimensions match. To determine result dimensions: for A + B, result has same dimensions as A and B; for AB where A is m×nm \times n and B is n×pn \times p, result is m×pm \times p. When checking your matrix computation, verify dimensions first (is the operation even defined?), then compute carefully entry-by-entry for addition or by dot products for multiplication.

Question 16

Let AA and BB be 2×32\times 3 matrices: A=[2−1034−2],B=[−3510−24].A=\begin{bmatrix}2&-1&0\\3&4&-2\end{bmatrix},\quad B=\begin{bmatrix}-3&5&1\\0&-2&4\end{bmatrix}. Given the matrices above, what is A+BA+B?

  1. [−141322]\begin{bmatrix}-1&4&1\\3&2&2\end{bmatrix} (correct answer)
  2. [5−6−136−6]\begin{bmatrix}5&-6&-1\\3&6&-6\end{bmatrix}
  3. [−14132−6]\begin{bmatrix}-1&4&1\\3&2&-6\end{bmatrix}
  4. Not defined (dimensions are incompatible).
Explanation: This question tests understanding of matrix addition. Matrix addition and subtraction are defined only when both matrices have identical dimensions, and the operation is performed entry-by-entry: (A±B)ij=Aij±Bij(A \pm B)_{ij} = A_{ij} \pm B_{ij}. To compute A + B, we add corresponding entries: the entry in row 1, column 1 of the result is obtained by adding the entry in row 1, column 1 of A with the same-position entry of B, giving 2+(−3)=−12 + (-3) = -1. Choice A is correct because it shows the computation with specific numbers: [−141322]\begin{bmatrix} -1 & 4 & 1 \\ 3 & 2 & 2 \end{bmatrix} from adding each pair like −1+5=4-1 + 5 = 4 and −2+4=2-2 + 4 = 2. Choice B makes an arithmetic error by subtracting instead of adding, computing A - B to get entries like 2−(−3)=52 - (-3) = 5 and −2−4=−6-2 - 4 = -6. Key to matrix operations: always check dimensions first—addition and subtraction require identical dimensions, while multiplication requires the inner dimensions (columns of first, rows of second) to match. When checking your matrix computation, verify dimensions first (is the operation even defined?), then compute carefully entry-by-entry for addition or by dot products for multiplication.

Question 17

Let AA and BB be matrices with dimensions AA is 2×32 \times 3 and BB is 2×22 \times 2: A=[1−20314],B=[210−3].A=\begin{bmatrix}1&-2&0\\3&1&4\end{bmatrix},\quad B=\begin{bmatrix}2&1\\0&-3\end{bmatrix}. Given the matrices above, why is the operation ABAB not defined?

  1. It is not defined because matrix multiplication requires the two matrices to have the same dimensions.
  2. It is not defined because the number of columns of AA (33) does not equal the number of rows of BB (22). (correct answer)
  3. It is not defined because AA has more rows than BB.
  4. It is not defined because AA contains a zero entry.
Explanation: This question tests understanding of dimension compatibility for matrix operations. For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix; the result has dimensions equal to the rows of the first and columns of the second. The operation AB requires the number of columns of A (33) to equal the number of rows of B (22), and since A is 2×32 \times 3 and B is 2×22 \times 2, the operation is not defined. Choice B is correct because it states the dimension requirement not met: columns of A (33) ≠\neq rows of B (22). Choice A incorrectly assumes the operation is defined despite dimension incompatibility: 2×32 \times 3 and 2×22 \times 2 cannot be multiplied because inner dimensions 3≠23 \neq 2. Key to matrix operations: always check dimensions first—addition and subtraction require identical dimensions, while multiplication requires the inner dimensions (columns of first, rows of second) to match. To determine result dimensions: for A + B, result has same dimensions as A and B; for AB where A is m×nm \times n and B is n×pn \times p, result is m×pm \times p.

Question 18

Let MM be a 2×22\times 2 matrix and I2I_2 the 2×22\times 2 identity matrix: M=[0−321],I2=[1001].M=\begin{bmatrix}0&-3\\2&1\end{bmatrix},\quad I_2=\begin{bmatrix}1&0\\0&1\end{bmatrix}. Based on the matrices defined, what is MI2MI_2?

  1. [1−322]\begin{bmatrix}1&-3\\2&2\end{bmatrix}
  2. [0−321]\begin{bmatrix}0&-3\\2&1\end{bmatrix} (correct answer)
  3. [0000]\begin{bmatrix}0&0\\0&0\end{bmatrix}
  4. [03−21]\begin{bmatrix}0&3\\-2&1\end{bmatrix}
Explanation: This question tests understanding of matrix multiplication. Matrix multiplication of an m×nm\times n matrix A with an n×pn\times p matrix B produces an m×pm\times p matrix AB, where entry (i,ji,j) equals the dot product of row i of A with column j of B. Since I_2 is the identity, MI_2 should equal M itself: [0−321]\begin{bmatrix} 0 & -3 \\ 2 & 1 \end{bmatrix}. Choice B is correct because multiplying by the identity leaves M unchanged: [0−321]\begin{bmatrix} 0 & -3 \\ 2 & 1 \end{bmatrix}. Choice D incorrectly has signs flipped in some entries, perhaps computing I_2 M instead or a sign error. For matrix multiplication, remember the row-column rule: entry (i,ji,j) of AB comes from multiplying corresponding entries of row i of A and column j of B, then summing those products. Unlike addition, matrix multiplication is not commutative (AB≠BAAB \neq BA in general), so always perform operations in the order specified.

Question 19

Let CC and DD be 2×32\times 3 matrices: C=[10−23−14],D=[−52103−4].C=\begin{bmatrix}1&0&-2\\3&-1&4\end{bmatrix},\quad D=\begin{bmatrix}-5&2&1\\0&3&-4\end{bmatrix}. Based on the matrices defined, what is C−DC-D?

  1. [−42−1320]\begin{bmatrix}-4&2&-1\\3&2&0\end{bmatrix}
  2. [6−2−33−48]\begin{bmatrix}6&-2&-3\\3&-4&8\end{bmatrix} (correct answer)
  3. [6−2−33−40]\begin{bmatrix}6&-2&-3\\3&-4&0\end{bmatrix}
  4. [−623−34−8]\begin{bmatrix}-6&2&3\\-3&4&-8\end{bmatrix}
Explanation: This question tests understanding of matrix subtraction. Matrix addition and subtraction are defined only when both matrices have identical dimensions, and the operation is performed entry-by-entry: (A ± B)ᵢⱼ = Aᵢⱼ ± Bᵢⱼ. To compute C - D, we subtract corresponding entries: the entry in row 1, column 1 of the result is obtained by subtracting the entry in row 1, column 1 of D from the same-position entry of C, giving 1 - (-5) = 6, and similarly for others. Choice B is correct because it shows the computation with specific numbers: [[1 - (-5), 0 - 2, -2 - 1],[3 - 0, -1 - 3, 4 - (-4)]] = [[6,-2,-3],[3,-4,8]]. Choice D has a sign error in the entry (1,1), likely from incorrectly subtracting as - (1 - (-5)) = -6 instead of 6. Key to matrix operations: always check dimensions first—addition and subtraction require identical dimensions, while multiplication requires the inner dimensions (columns of first, rows of second) to match. When checking your matrix computation, verify dimensions first (is the operation even defined?), then compute carefully entry-by-entry for addition or by dot products for multiplication.

Question 20

Let AA and BB be 2×22\times 2 matrices: A=[1234],B=[01−12].A=\begin{bmatrix}1&2\\3&4\end{bmatrix},\quad B=\begin{bmatrix}0&1\\-1&2\end{bmatrix}. Given the matrices above, which statement about ABAB and BABA is true?

  1. AB=BAAB=BA for all 2×22\times 2 matrices.
  2. ABAB is defined but BABA is not defined.
  3. AB≠BAAB\ne BA for these matrices. (correct answer)
  4. ABAB and BABA are both undefined because matrix multiplication requires equal matrices.
Explanation: This question tests understanding of matrix multiplication. Unlike real number multiplication, matrix multiplication is not commutative: in general, AB≠BAAB \neq BA, even when both products are defined. Computing AB gives [−25−411]\begin{bmatrix} -2 & 5 \\ -4 & 11 \end{bmatrix} while computing BA gives [3456]\begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix}, demonstrating that matrix multiplication depends on the order of the factors. Choice C is correct because it shows the reasoning with specific numbers: AB≠BAAB \neq BA for these matrices. Choice A assumes AB=BAAB = BA, but matrix multiplication is not commutative, so the order matters. Unlike addition, matrix multiplication is not commutative (AB≠BAAB \neq BA in general), so always perform operations in the order specified. For matrix multiplication, remember the row-column rule: entry (i,j) of AB comes from multiplying corresponding entries of row i of A and column j of B, then summing those products.