AP Precalculus Flashcards: Matrices

What this deck covers

This deck focuses on Matrices, giving you a quick way to review the definitions, rules, and examples that matter most for AP 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: Which property describes $AB = BA$ for matrices $A$ and $B$?

Answer: Commutative property (rarely holds for matrices). Matrix multiplication is generally not commutative.

Flashcard 2: Can you multiply a $2 \times 3$ matrix by a $3 \times 2$ matrix?

Answer: Yes, resulting in a $2 \times 2$ matrix. Inner dimensions match (3), so multiplication is valid.

Flashcard 3: What is a column matrix?

Answer: A matrix with a single column. Also called a column vector with dimension $m×1$.

Flashcard 4: Find the result of $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$.

Answer: $\begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$. Identity matrix preserves the second matrix unchanged.

Flashcard 5: How do you denote the transpose of a matrix $A$?

Answer: $A^T$. Standard notation for transpose operation.

Flashcard 6: What is the determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

Answer: $ad - bc$. Formula for $2×2$ determinant calculation.

Flashcard 7: What is the result of multiplying any matrix by a zero matrix?

Answer: A zero matrix. Zero matrix absorbs any matrix in multiplication.

Flashcard 8: Define the term 'symmetric matrix'.

Answer: A matrix equal to its transpose, $A = A^T$. Matrix equals its own transpose.

Flashcard 9: Identify the inverse of a matrix $A$ if it exists.

Answer: Matrix $A^{-1}$ such that $AA^{-1} = I$. Inverse satisfies the multiplicative identity property.

Flashcard 10: What is the determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

Answer: $ad - bc$. Formula for $2×2$ determinant calculation.

Flashcard 11: State the dimensions of a $3 \times 4$ matrix.

Answer: 3 rows and 4 columns. Matrix dimensions are always written as rows × columns.

Flashcard 12: What is a zero matrix?

Answer: A matrix where all elements are zero. The additive identity in matrix operations.

Flashcard 13: Find the trace of matrix $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$.

Answer:

1. Trace equals sum of diagonal elements: $1+4=5$.

Flashcard 14: What is the identity element for matrix addition?

Answer: The zero matrix. Adding zero matrix leaves any matrix unchanged.

Flashcard 15: What is a diagonal matrix?

Answer: A matrix where non-diagonal elements are zero. Only diagonal entries can be non-zero.

Flashcard 16: Calculate the sum of matrices $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\begin{pmatrix} 4 & 3 \\ 2 & 1 \end{pmatrix}$.

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

Flashcard 17: Find the result of $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$.

Answer: $\begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$. Identity matrix preserves the second matrix unchanged.

Flashcard 18: What does it mean for two matrices to be equal?

Answer: All corresponding entries are equal. Element-by-element equality for same dimensions.

Flashcard 19: What condition must two matrices meet to be added together?

Answer: They must have the same dimensions. Addition is defined element-wise for matrices.

Flashcard 20: What is the inverse of the identity matrix $I$?

Answer: The identity matrix $I$ itself. Identity matrix is its own inverse: $I^{-1} = I$.

Flashcard 21: Calculate the determinant of $\begin{pmatrix} 5 & 2 \\ 3 & 7 \end{pmatrix}$.

Answer:

1. Using formula $ad - bc = 5(7) - 2(3) = 35 - 6$.

Flashcard 22: How do you denote the transpose of a matrix $A$?

Answer: $A^T$. Standard notation for transpose operation.

Flashcard 23: What is the result of transposing a column matrix?

Answer: A row matrix. Transpose flips dimensions from column to row.

Flashcard 24: What is a matrix?

Answer: A rectangular array of numbers arranged in rows and columns. The fundamental definition of a matrix structure.

Flashcard 25: Identify the inverse of a matrix $A$ if it exists.

Answer: Matrix $A^{-1}$ such that $AA^{-1} = I$. Inverse satisfies the multiplicative identity property.

Flashcard 26: How is the element in the second row, first column of a matrix denoted?

Answer: $a_{21}$. Subscript notation: first index is row, second is column.

Flashcard 27: What is the inverse of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

Answer: $\frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ if $ad-bc \neq 0$. Formula for $2×2$ matrix inverse when determinant ≠ 0.

Flashcard 28: Check if $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is invertible.

Answer: Invertible, since determinant $\neq 0$. Determinant $= 1(4) - 2(3) = -2 ≠ 0$.

Flashcard 29: What is a diagonal matrix?

Answer: A matrix where non-diagonal elements are zero. Only diagonal entries can be non-zero.

Flashcard 30: What is the result of multiplying a matrix by the identity matrix?

Answer: The original matrix. Identity matrix leaves any matrix unchanged when multiplied.

Flashcard 31: How is the element in the second row, first column of a matrix denoted?

Answer: $a_{21}$. Subscript notation: first index is row, second is column.

Flashcard 32: What is the result of multiplying a matrix by the identity matrix?

Answer: The original matrix. Identity matrix leaves any matrix unchanged when multiplied.

Flashcard 33: If $A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, what type of matrix is $A$?

Answer: Skew-symmetric matrix. Since $A^T = -A$, it satisfies skew-symmetric property.

Flashcard 34: What is the associative property of matrix multiplication?

Answer: $(AB)C = A(BC)$. Matrix multiplication groups from either direction.

Flashcard 35: State the dimensions of a $3 \times 4$ matrix.

Answer: 3 rows and 4 columns. Matrix dimensions are always written as rows × columns.

Flashcard 36: What is a column matrix?

Answer: A matrix with a single column. Also called a column vector with dimension $m×1$.

Flashcard 37: Check if $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is invertible.

Answer: Invertible, since determinant $\neq 0$. Determinant $= 1(4) - 2(3) = -2 ≠ 0$.

Flashcard 38: What is a zero matrix?

Answer: A matrix where all elements are zero. The additive identity in matrix operations.

Flashcard 39: What is the trace of a matrix?

Answer: The sum of the diagonal elements. Sum of elements along the main diagonal.

Flashcard 40: What is a row matrix?

Answer: A matrix with a single row. Also called a row vector with dimension $1×n$.

Flashcard 41: Define the term 'symmetric matrix'.

Answer: A matrix equal to its transpose, $A = A^T$. Matrix equals its own transpose.

Flashcard 42: What is the rank of a matrix?

Answer: The maximum number of linearly independent row or column vectors. Measures the dimension of the row or column space.

Flashcard 43: If $A$ is a $3 \times 2$ matrix, what are the dimensions of $A^T$?

Answer: $2 \times 3$. Transpose flips dimensions: rows become columns.

Flashcard 44: What is a row matrix?

Answer: A matrix with a single row. Also called a row vector with dimension $1×n$.

Flashcard 45: Determine the dimensions of zero matrix $O = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$.

Answer: $2 \times 3$. Matrix has 2 rows and 3 columns of zeros.

Flashcard 46: What is a skew-symmetric matrix?

Answer: A matrix where $A^T = -A$. Transpose equals negative of original matrix.

Flashcard 47: What condition must two matrices meet to be added together?

Answer: They must have the same dimensions. Addition is defined element-wise for matrices.

Flashcard 48: What is a matrix?

Answer: A rectangular array of numbers arranged in rows and columns. The fundamental definition of a matrix structure.

Flashcard 49: What is the rank of a matrix?

Answer: The maximum number of linearly independent row or column vectors. Measures the dimension of the row or column space.

Flashcard 50: Define an identity matrix.

Answer: A square matrix with 1s on the main diagonal and 0s elsewhere. The multiplicative identity in matrix operations.

Flashcard 51: Identify the main diagonal elements in a $3 \times 3$ matrix.

Answer: Elements $a_{11}$, $a_{22}$, $a_{33}$. Main diagonal runs from top-left to bottom-right.

Flashcard 52: What operation is performed by $2A$ for a matrix $A$?

Answer: Scalar multiplication, doubling each element. Multiplies each matrix element by the scalar.

Flashcard 53: What operation is performed by $2A$ for a matrix $A$?

Answer: Scalar multiplication, doubling each element. Multiplies each matrix element by the scalar.

Flashcard 54: What is the scalar multiplication of $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ by 3?

Answer: $\begin{pmatrix} 3 & 6 \\ 9 & 12 \end{pmatrix}$. Multiply each element by 3.

Flashcard 55: What is a square matrix?

Answer: A matrix with the same number of rows and columns. Equal row and column counts define a square matrix.

Flashcard 56: Define an identity matrix.

Answer: A square matrix with 1s on the main diagonal and 0s elsewhere. The multiplicative identity in matrix operations.

Flashcard 57: Which property describes $AB = BA$ for matrices $A$ and $B$?

Answer: Commutative property (rarely holds for matrices). Matrix multiplication is generally not commutative.

Flashcard 58: Calculate the sum of matrices $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\begin{pmatrix} 4 & 3 \\ 2 & 1 \end{pmatrix}$.

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

Flashcard 59: What does it mean for two matrices to be equal?

Answer: All corresponding entries are equal. Element-by-element equality for same dimensions.