ACT Math 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 ACT Math.

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: Identify the order of a $3 \times 2$ matrix.

Answer: 3 rows and 2 columns. The first number is rows, second is columns.

Flashcard 2: What is an invertible matrix?

Answer: A square matrix with a non-zero determinant. Non-zero determinant means the matrix has an inverse.

Flashcard 3: Define a zero matrix.

Answer: A matrix where all elements are zero. Acts as the additive identity in matrix operations.

Flashcard 4: What does the identity matrix do in multiplication?

Answer: Leaves any matrix unchanged when multiplied. Multiplicative identity property: $AI = IA = A$.

Flashcard 5: What is an invertible matrix?

Answer: A square matrix with a non-zero determinant. Non-zero determinant means the matrix has an inverse.

Flashcard 6: Find the transpose of $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$.

Answer: $\begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$. Each row becomes a column in the transposed matrix.

Flashcard 7: What is matrix addition?

Answer: Adding corresponding elements of two matrices of the same order. Both matrices must have identical dimensions.

Flashcard 8: What is the condition for matrix multiplication to be defined?

Answer: Number of columns in the first matrix must equal the number of rows in the second. Inner dimensions must match for multiplication to work.

Flashcard 9: What is a symmetric matrix?

Answer: A matrix that is equal to its transpose. Mirror symmetry across the main diagonal.

Flashcard 10: What does it mean if two matrices are equal?

Answer: They have the same dimensions and corresponding elements are equal. Same size and all corresponding entries match exactly.

Flashcard 11: What is a transpose of a matrix?

Answer: Switching the rows and columns of a matrix. Rows become columns and columns become rows.

Flashcard 12: What is an orthogonal matrix?

Answer: A square matrix whose transpose is also its inverse. Special property where $A^T = A^{-1}$.

Flashcard 13: Find the trace of $\begin{bmatrix} 3 & 0 \ 0 & 7 \end{bmatrix}$.

Answer:

1. Sum the diagonal elements: $3 + 7 = 10$.

Flashcard 14: What is the order of a matrix?

Answer: The dimensions of a matrix, given as rows $\times$ columns. Specifies the size and shape of the matrix.

Flashcard 15: Define the inverse of a matrix.

Answer: A matrix that, when multiplied with the original, results in the identity matrix. Denoted as $A^{-1}$ where $A \cdot A^{-1} = I$.

Flashcard 16: What is an orthogonal matrix?

Answer: A square matrix whose transpose is also its inverse. Special property where $A^T = A^{-1}$.

Flashcard 17: What is a diagonal matrix?

Answer: A square matrix where all off-diagonal elements are zero. Non-zero elements only appear along the main diagonal.

Flashcard 18: What is matrix subtraction?

Answer: Subtracting corresponding elements of two matrices of the same order. Subtract element-wise from corresponding positions.

Flashcard 19: Calculate $2 \times \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$.

Answer: $\begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}$. Each element multiplied by $2$: $(2 \cdot 1, 2 \cdot 2, 2 \cdot 3, 2 \cdot 4)$.

Flashcard 20: Define a zero matrix.

Answer: A matrix where all elements are zero. Acts as the additive identity in matrix operations.

Flashcard 21: How do you denote a matrix?

Answer: Typically with uppercase letters, e.g., $A$, $B$, $C$. Standard mathematical convention for matrix notation.

Flashcard 22: What is the condition for matrix multiplication to be defined?

Answer: Number of columns in the first matrix must equal the number of rows in the second. Inner dimensions must match for multiplication to work.

Flashcard 23: What is a column matrix?

Answer: A matrix with a single column. Also called a column vector in linear algebra.

Flashcard 24: What is the determinant of $\begin{bmatrix} a & b \ c & d \end{bmatrix}$?

Answer: $ad - bc$. Formula for $2 \times 2$ matrix determinant calculation.

Flashcard 25: Calculate the sum of $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $\begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}$.

Answer: $\begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$. Add element-wise: $(1+5, 2+6, 3+7, 4+8)$.

Flashcard 26: What is the order of a matrix?

Answer: The dimensions of a matrix, given as rows $\times$ columns. Specifies the size and shape of the matrix.

Flashcard 27: Is $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ symmetric?

Answer: No. Not symmetric because $a_{12} \neq a_{21}$ (2 ≠ 3).

Flashcard 28: What is an identity matrix?

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

Flashcard 29: What is matrix subtraction?

Answer: Subtracting corresponding elements of two matrices of the same order. Subtract element-wise from corresponding positions.

Flashcard 30: What does the identity matrix do in multiplication?

Answer: Leaves any matrix unchanged when multiplied. Multiplicative identity property: $AI = IA = A$.

Flashcard 31: Find the transpose of $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$

Answer: $\begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$. Each row becomes a column in the transposed matrix.

Flashcard 32: What is a singular matrix?

Answer: A square matrix with a determinant of zero. Cannot be inverted due to zero determinant.

Flashcard 33: What is the trace of a matrix?

Answer: The sum of the diagonal elements of a square matrix. Add all diagonal elements from top-left to bottom-right.

Flashcard 34: What is a diagonal matrix?

Answer: A square matrix where all off-diagonal elements are zero. Non-zero elements only appear along the main diagonal.

Flashcard 35: What is a symmetric matrix?

Answer: A matrix that is equal to its transpose. Mirror symmetry across the main diagonal.

Flashcard 36: What is scalar multiplication in matrices?

Answer: Multiplying each element of a matrix by a scalar. Multiply every element by the same scalar value.

Flashcard 37: What is the trace of a matrix?

Answer: The sum of the diagonal elements of a square matrix. Add all diagonal elements from top-left to bottom-right.

Flashcard 38: What is a determinant?

Answer: A scalar value that is a function of a square matrix. Measures certain properties of square matrices.

Flashcard 39: What is an identity matrix?

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

Flashcard 40: What is a row matrix?

Answer: A matrix with a single row. Also called a row vector in linear algebra.

Flashcard 41: Calculate the sum of $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $\begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}$.

Answer: $\begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$. Add element-wise: $(1+5, 2+6, 3+7, 4+8)$.

Flashcard 42: What is a matrix?

Answer: A rectangular array of numbers arranged in rows and columns. Numbers organized in a grid format for mathematical operations.

Flashcard 43: Which matrix operation is not commutative?

Answer: Matrix multiplication. Order matters in matrix multiplication: $AB \neq BA$.

Flashcard 44: Is $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ symmetric?

Answer: No. Not symmetric because $a_{12} \neq a_{21}$ (2 ≠ 3).

Flashcard 45: What is a square matrix?

Answer: A matrix with the same number of rows and columns. Equal dimensions create a special matrix type.

Flashcard 46: Calculate $2 \times \begin{bmatrix} 1 & 2 \ 3 & 4 \ \end{bmatrix}$

Answer: $\begin{bmatrix} 2 & 4 \ 6 & 8 \ \end{bmatrix}$. Each element multiplied by $2$: $(2 \cdot 1, 2 \cdot 2, 2 \cdot 3, 2 \cdot 4)$

Flashcard 47: What is the determinant of $\begin{bmatrix} a & b \ c & d \end{bmatrix}$?

Answer: $ad - bc$. Formula for $2 \times 2$ matrix determinant calculation.

Flashcard 48: What is a column matrix?

Answer: A matrix with a single column. Also called a column vector in linear algebra.

Flashcard 49: What is a row matrix?

Answer: A matrix with a single row. Also called a row vector in linear algebra.

Flashcard 50: What does it mean if two matrices are equal?

Answer: They have the same dimensions and corresponding elements are equal. Same size and all corresponding entries match exactly.

Flashcard 51: What is a transpose of a matrix?

Answer: Switching the rows and columns of a matrix. Rows become columns and columns become rows.

Flashcard 52: Identify the order of a $3 \times 2$ matrix.

Answer: 3 rows and 2 columns. The first number is rows, second is columns.

Flashcard 53: What is a matrix?

Answer: A rectangular array of numbers arranged in rows and columns. Numbers organized in a grid format for mathematical operations.

Flashcard 54: What is matrix multiplication?

Answer: Dot product of rows and columns from two matrices. Row elements multiplied by column elements, then summed.

Flashcard 55: What is a square matrix?

Answer: A matrix with the same number of rows and columns. Equal dimensions create a special matrix type.

Flashcard 56: What is a determinant?

Answer: A scalar value that is a function of a square matrix. Measures certain properties of square matrices.

Flashcard 57: Define the inverse of a matrix.

Answer: A matrix that, when multiplied with the original, results in the identity matrix. Denoted as $A^{-1}$ where $A \cdot A^{-1} = I$.

Flashcard 58: Which matrix operation is not commutative?

Answer: Matrix multiplication. Order matters in matrix multiplication: $AB \neq BA$.

Flashcard 59: What is matrix multiplication?

Answer: Dot product of rows and columns from two matrices. Row elements multiplied by column elements, then summed.

Flashcard 60: What is matrix addition?

Answer: Adding corresponding elements of two matrices of the same order. Both matrices must have identical dimensions.

Flashcard 61: Find the trace of $\begin{bmatrix} 3 & 0 \ 0 & 7 \end{bmatrix}$.

Answer:

1. Sum the diagonal elements: $3 + 7 = 10$.

Flashcard 62: How do you denote a matrix?

Answer: Typically with uppercase letters, e.g., $A$, $B$, $C$. Standard mathematical convention for matrix notation.

Flashcard 63: What is scalar multiplication in matrices?

Answer: Multiplying each element of a matrix by a scalar. Multiply every element by the same scalar value.