Precalculus Flashcards: Matrix Addition Subtraction And Multiplication

What this deck covers

This deck focuses on Matrix Addition Subtraction And Multiplication, giving you a quick way to review the definitions, rules, and examples that matter most for Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is $A+B$ for $A=\begin{pmatrix}1&-2\\3&0\end{pmatrix}$ and $B=\begin{pmatrix}4&5\\-1&2\end{pmatrix}$?

Answer: $\begin{pmatrix}5&3\\2&2\end{pmatrix}$. Add corresponding entries: $(1+4, -2+5; 3+(-1), 0+2)$.

Flashcard 2: What is $3A$ if $A=\begin{bmatrix}-1&2\\0&5\end{bmatrix}$?

Answer: $\begin{bmatrix}-3&6\\0&15\end{bmatrix}$. Multiply each entry by 3: $3(-1)=-3$, $3(2)=6$, $3(0)=0$, $3(5)=15$.

Flashcard 3: What is $A+B$ if $A=\begin{bmatrix}1&-2\\3&0\end{bmatrix}$ and $B=\begin{bmatrix}4&5\\-1&2\end{bmatrix}$?

Answer: $\begin{bmatrix}5&3\\2&2\end{bmatrix}$. Add corresponding entries: $1+4=5$, $-2+5=3$, $3+(-1)=2$, $0+2=2$.

Flashcard 4: Which statement is true about commutativity of matrix multiplication: $AB=BA$ always?

Answer: False; in general $AB\ne BA$ (and one product may be undefined). Matrix multiplication is not commutative.

Flashcard 5: What condition on dimensions must be true to subtract matrices $A-B$?

Answer: $A$ and $B$ must have the same dimensions $m \times n$. Subtraction requires matching rows and columns to subtract corresponding entries.

Flashcard 6: What is the required dimension condition for the product $AB$ to be defined?

Answer: If $A$ is $m\times n$ and $B$ is $n\times p$. Number of columns in $A$ must equal number of rows in $B$.

Flashcard 7: What is the multiplicative identity for an $n\times n$ matrix $A$?

Answer: The identity matrix $I_n$, so $AI_n=I_nA=A$. Identity matrix leaves any square matrix unchanged.

Flashcard 8: What is $(AB)C$ compared to $A(BC)$ when all products are defined?

Answer: $(AB)C=A(BC)$. Matrix multiplication is associative.

Flashcard 9: What is the entrywise rule for matrix subtraction $(A-B)_{ij}$?

Answer: $(A-B)_{ij}=a_{ij}-b_{ij}$. Subtract corresponding entries at the same position in both matrices.

Flashcard 10: What is the identity matrix property for multiplication with $I_n$?

Answer: $I_nA=A$ and $AI_n=A$ (when dimensions match). Identity matrix acts as multiplicative identity element.

Flashcard 11: What condition on dimensions is required to add matrices $A$ and $B$?

Answer: $A$ and $B$ must have the same dimensions ($m\times n$). Addition requires corresponding entries to exist.

Flashcard 12: What is $AB$ if $A=\begin{bmatrix}1\\-2\\3\end{bmatrix}$ and $B=\begin{bmatrix}4&0\end{bmatrix}$?

Answer: $\begin{bmatrix}4&0\\-8&0\\12&0\end{bmatrix}$. Column times row gives outer product matrix.

Flashcard 13: What is the dot-product formula for an entry $(AB)_{ij}$?

Answer: $(AB)_{ij}=\sum_{k=1}^{n} a_{ik}b_{kj}$. Row $i$ of $A$ dot product with column $j$ of $B$.

Flashcard 14: What is $AB$ if $A=\begin{bmatrix}2&-1&0\end{bmatrix}$ and $B=\begin{bmatrix}3\\4\\5\end{bmatrix}$?

Answer: $2$. Row vector times column vector gives scalar: $2(3)+(-1)(4)+0(5)=2$.

Flashcard 15: What is the scalar multiplication rule for entries of $cA$?

Answer: $(cA)_{ij}=c\,a_{ij}$. Multiply each entry by the scalar $c$.

Flashcard 16: What is the entrywise rule for matrix addition $(A+B)_{ij}$?

Answer: $(A+B)_{ij}=a_{ij}+b_{ij}$. Add corresponding entries at the same position in both matrices.

Flashcard 17: What is $A-B$ if $A=\begin{bmatrix}6&1\\-2&4\end{bmatrix}$ and $B=\begin{bmatrix}3&7\\5&-1\end{bmatrix}$?

Answer: $\begin{bmatrix}3&-6\\-7&5\end{bmatrix}$. Subtract corresponding entries: $6-3=3$, $1-7=-6$, $-2-5=-7$, $4-(-1)=5$.

Flashcard 18: What is $AB$ for $A=\begin{pmatrix}1&0\\-2&3\end{pmatrix}$ and $B=\begin{pmatrix}5\\-1\end{pmatrix}$?

Answer: $\begin{pmatrix}5\\-13\end{pmatrix}$. Row 1: $(1)(5)+(0)(-1)=5$; Row 2: $(-2)(5)+(3)(-1)=-10-3=-13$.

Flashcard 19: What is $AB$ for $A=\begin{pmatrix}2&-1&0\end{pmatrix}$ and $B=\begin{pmatrix}3\\4\\-2\end{pmatrix}$?

Answer: $2$. $(2)(3)+(-1)(4)+(0)(-2)=6-4+0=2$.

Flashcard 20: What is the additive identity matrix for addition with an $m\times n$ matrix $A$?

Answer: The $m\times n$ zero matrix $0$, so $A+0=A$. Adding zero matrix leaves any matrix unchanged.

Flashcard 21: What condition on dimensions must be true to multiply matrices $AB$?

Answer: If $A$ is $m\times n$, then $B$ must be $n\times p$. Number of columns in $A$ must equal number of rows in $B$.

Flashcard 22: What is $A+(-A)$ for any matrix $A$?

Answer: The zero matrix of the same size as $A$. A matrix plus its negative gives all zero entries.

Flashcard 23: What is the distributive property of matrix multiplication over addition?

Answer: $A(B+C)=AB+AC$ and $(A+B)C=AC+BC$. Multiplication distributes over addition from both sides.

Flashcard 24: If $A$ is $m\times n$ and $B$ is $n\times p$, what is the size of $AB$?

Answer: $AB$ is $m\times p$. Product has rows of $A$ and columns of $B$.

Flashcard 25: What is $AB$ for $A=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ and $B=\begin{pmatrix}0&1\\-1&2\end{pmatrix}$?

Answer: $\begin{pmatrix}-2&5\\-4&11\end{pmatrix}$. Row 1: $(1)(0)+(2)(-1)=-2$, $(1)(1)+(2)(2)=5$; Row 2: $(3)(0)+(4)(-1)=-4$, $(3)(1)+(4)(2)=11$.

Flashcard 26: What is $AB$ if $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ and $B=\begin{bmatrix}0&1\\-1&2\end{bmatrix}$?

Answer: $\begin{bmatrix}-2&5\\-4&11\end{bmatrix}$. Row-column products: $(1,2)\cdot(0,-1)=-2$, $(1,2)\cdot(1,2)=5$, etc.

Flashcard 27: What is $I_2\begin{bmatrix}a&b\\c&d\end{bmatrix}$?

Answer: $\begin{bmatrix}a&b\\c&d\end{bmatrix}$. Identity matrix preserves all entries when multiplied.

Flashcard 28: What condition on dimensions must be true to add matrices $A$ and $B$?

Answer: $A$ and $B$ must have the same dimensions $m \times n$. Addition requires matching rows and columns to add corresponding entries.

Flashcard 29: What is the size (dimensions) of $AB$ if $A$ is $m\times n$ and $B$ is $n\times p$?

Answer: $AB$ is $m\times p$. Product has $A$'s rows and $B$'s columns.

Flashcard 30: What distributive property relates multiplication over addition: $A(B+C)$?

Answer: $A(B+C)=AB+AC$ (when dimensions are compatible). Matrix multiplication distributes over addition.

Flashcard 31: What is the formula for an entry of a product matrix $(AB)_{ij}$?

Answer: $(AB)_{ij}=\sum_{k=1}^{n} a_{ik}b_{kj}$. Dot product of row $i$ of $A$ with column $j$ of $B$.

Flashcard 32: What is the commutative property status for matrix multiplication?

Answer: In general, $AB\ne BA$ even when both products exist. Matrix multiplication is not commutative unlike scalar multiplication.

Flashcard 33: What is $A-B$ for $A=\begin{pmatrix}7&1\\-3&4\end{pmatrix}$ and $B=\begin{pmatrix}2&-5\\6&0\end{pmatrix}$?

Answer: $\begin{pmatrix}5&6\\-9&4\end{pmatrix}$. Subtract corresponding entries: $(7-2, 1-(-5); -3-6, 4-0)$.