AP Precalculus Flashcards: Matrices As Functions

What this deck covers

This deck focuses on Matrices As Functions, 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: Define a square matrix.

Answer: A matrix with the same number of rows and columns. Equal dimensions make it square-shaped.

Flashcard 2: Identify the transpose of matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

Answer: $A^T = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$. Transpose switches rows and columns.

Flashcard 3: Define zero matrix.

Answer: A matrix in which all elements are zero. All entries equal zero.

Flashcard 4: What is the commutative property of matrix multiplication?

Answer: Matrix multiplication is not commutative. Generally $AB ≠ BA$ for matrices.

Flashcard 5: State the order of a matrix with 3 rows and 4 columns.

Answer: The order is $3 \times 4$. Order is written as rows × columns.

Flashcard 6: What is the result of $A + 0$ for any matrix $A$?

Answer: The matrix $A$ itself. Zero matrix is additive identity.

Flashcard 7: What condition makes a $2 \times 2$ matrix singular?

Answer: The determinant is zero. Zero determinant means no inverse exists.

Flashcard 8: What is matrix addition?

Answer: The sum of two matrices by adding corresponding elements. Add elements in same positions.

Flashcard 9: What is the element in the second row, third column of matrix $A$?

Answer: $a_{2,3}$. Standard notation: $a_{row,column}$.

Flashcard 10: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, find $A - A$.

Answer: $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$. Subtracting gives the zero matrix.

Flashcard 11: What condition is necessary for matrix multiplication $AB$?

Answer: The number of columns in $A$ must equal the number of rows in $B$. Inner dimensions must match for multiplication.

Flashcard 12: What is the identity matrix of order 2?

Answer: $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Diagonal of ones, zeros elsewhere.

Flashcard 13: What is a matrix?

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

Flashcard 14: Find the determinant of $\begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix}$.

Answer: The determinant is $7$. Calculate: $(3)(5) - (4)(2) = 15 - 8 = 7$.

Flashcard 15: Define a square matrix.

Answer: A matrix with the same number of rows and columns. Equal dimensions make it square-shaped.

Flashcard 16: What is matrix multiplication?

Answer: The product of matrices by row-by-column multiplication. Dot product of rows and columns.

Flashcard 17: What is a symmetric matrix?

Answer: A matrix where $A = A^T$. Matrix equals its own transpose.

Flashcard 18: Find the trace of $\begin{bmatrix} 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6 \end{bmatrix}$.

Answer: The trace is $15$. Add diagonal elements: $4 + 5 + 6 = 15$.

Flashcard 19: What is a matrix?

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

Flashcard 20: What is the associative property of matrix multiplication?

Answer: $(AB)C = A(BC)$. Grouping doesn't affect the product.

Flashcard 21: What is the inverse of a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$?

Answer: $A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$. Formula for $2 \times 2$ matrix inverse.

Flashcard 22: What is the element in the second row, third column of matrix $A$?

Answer: $a_{2,3}$. Standard notation: $a_{row,column}$.

Flashcard 23: What is the trace of a matrix?

Answer: The sum of the diagonal elements of a square matrix. Sum of main diagonal elements.

Flashcard 24: What is matrix multiplication?

Answer: The product of matrices by row-by-column multiplication. Dot product of rows and columns.

Flashcard 25: Is $\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ symmetric?

Answer: Yes, it is symmetric. Check if matrix equals its transpose.

Flashcard 26: What is a diagonal matrix?

Answer: A matrix where all off-diagonal elements are zero. Non-diagonal entries are all zero.

Flashcard 27: What is a column matrix?

Answer: A matrix with only one column. Single vertical array of elements.

Flashcard 28: Define scalar multiplication of a matrix.

Answer: Multiplying each element of a matrix by a scalar. Scale each entry by the scalar value.

Flashcard 29: What is the identity property of matrix multiplication?

Answer: $AI = IA = A$, where $I$ is the identity matrix. Identity matrix preserves multiplication.

Flashcard 30: Find the determinant of $\begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix}$.

Answer: The determinant is $7$. Calculate: $(3)(5) - (4)(2) = 15 - 8 = 7$.

Flashcard 31: What is a row matrix?

Answer: A matrix with only one row. Single horizontal array of elements.

Flashcard 32: State the order of a matrix with 3 rows and 4 columns.

Answer: The order is $3 \times 4$. Order is written as rows × columns.

Flashcard 33: Define zero matrix.

Answer: A matrix in which all elements are zero. All entries equal zero.

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

Answer: $AI = IA = A$, where $I$ is the identity matrix. Identity matrix preserves multiplication.

Flashcard 35: What is the inverse of a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$?

Answer: $A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$. Formula for 2×2 matrix inverse.

Flashcard 36: What condition is necessary for two matrices to be added?

Answer: They must have the same dimensions. Matrices must be same size to add.

Flashcard 37: What is the trace of a matrix?

Answer: The sum of the diagonal elements of a square matrix. Sum of main diagonal elements.

Flashcard 38: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, what is $2A$?

Answer: $2A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$. Multiply each element by the scalar 2.

Flashcard 39: What is the associative property of matrix multiplication?

Answer: $(AB)C = A(BC)$. Grouping doesn't affect the product.

Flashcard 40: What is the determinant of a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$?

Answer: $ad - bc$. Cross-multiply and subtract for 2×2.

Flashcard 41: What is the result of $A + 0$ for any matrix $A$?

Answer: The matrix $A$ itself. Zero matrix is additive identity.

Flashcard 42: Find the trace of $\begin{bmatrix} 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6 \end{bmatrix}$.

Answer: The trace is $15$. Add diagonal elements: $4 + 5 + 6 = 15$.

Flashcard 43: What is an invertible matrix?

Answer: A matrix with a non-zero determinant and an inverse. Has an inverse that undoes multiplication.

Flashcard 44: What is a row matrix?

Answer: A matrix with only one row. Single horizontal array of elements.

Flashcard 45: What is an antisymmetric matrix?

Answer: A matrix where $A = -A^T$. Matrix equals negative of its transpose.

Flashcard 46: What is the identity matrix of order 2?

Answer: $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Diagonal of ones, zeros elsewhere.

Flashcard 47: Is $\begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix}$ antisymmetric?

Answer: Yes, it is antisymmetric. Check if matrix equals negative transpose.