ACT Math Matrices: Basic Operations and Applications

Matrices appear on the ACT but not on the SAT, so many students skip this topic. That's a mistake — matrix questions are usually easy points once you learn the rules.

What Is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. The size is described as "rows × columns."

This is a $2 \times 3$ matrix (2 rows, 3 columns).

Matrix Addition and Subtraction

Add or subtract corresponding entries. Both matrices must be the same size.

Example 1: $\begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix} + \begin{bmatrix} 1 & 4 \\ 0 & 3 \end{bmatrix} = \begin{bmatrix} 4 & 5 \\ 5 & 5 \end{bmatrix}$

Example 2: $\begin{bmatrix} 7 & 3 \\ 8 & 1 \end{bmatrix} - \begin{bmatrix} 2 & 5 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} 5 & -2 \\ 5 & 0 \end{bmatrix}$

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Scalar Multiplication

Multiply every entry by the scalar (a single number).

Example 3: $3 \begin{bmatrix} 2 & -1 \\ 0 & 4 \end{bmatrix} = \begin{bmatrix} 6 & -3 \\ 0 & 12 \end{bmatrix}$

Matrix Multiplication

This is where students get nervous, but the pattern is simple: row × column, then add.

To multiply $A \times B$ :

The number of columns in A must equal the number of rows in B
Result size: (rows of A) × (columns of B)

Example 4: $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \times \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$

Top-left: $(1)(5) + (2)(7) = 5 + 14 = 19$
Top-right: $(1)(6) + (2)(8) = 6 + 16 = 22$
Bottom-left: $(3)(5) + (4)(7) = 15 + 28 = 43$
Bottom-right: $(3)(6) + (4)(8) = 18 + 32 = 50$

The Identity Matrix

The identity matrix $I$ is the matrix equivalent of the number 1. For $2 \times 2$ :

Any matrix multiplied by $I$ stays the same: $AI = A$ .

Determinant of a 2×2 Matrix

For $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ :

Example 5: $\det\begin{bmatrix} 3 & 2 \\ 1 & 5 \end{bmatrix} = (3)(5) - (2)(1) = 15 - 2 = 13$

If the determinant is 0, the matrix has no inverse.

Real-World Applications

Example 6: A store sells 2 types of smoothies. Monday's sales: 10 small, 8 large. Tuesday's sales: 12 small, 6 large. Small costs $4, large costs$ 6.

Sales matrix: $S = \begin{bmatrix} 10 & 8 \\ 12 & 6 \end{bmatrix}$

Price matrix: $P = \begin{bmatrix} 4 \\ 6 \end{bmatrix}$

Revenue: $S \times P = \begin{bmatrix} 10(4) + 8(6) \\ 12(4) + 6(6) \end{bmatrix} = \begin{bmatrix} 88 \\ 84 \end{bmatrix}$

Monday revenue: $\$ 88 $, Tuesday revenue:$ $84$ .

Practice Problems

Problem 1: Calculate $2\begin{bmatrix} 4 & -1 \\ 3 & 2 \end{bmatrix} - \begin{bmatrix} 1 & 3 \\ 0 & -2 \end{bmatrix}$

Solution

$\begin{bmatrix} 8 & -2 \\ 6 & 4 \end{bmatrix} - \begin{bmatrix} 1 & 3 \\ 0 & -2 \end{bmatrix} = \begin{bmatrix} 7 & -5 \\ 6 & 6 \end{bmatrix}$

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

Solution

$\det = (5)(4) - (3)(2) = 20 - 6 = 14$

Problem 3: Multiply $\begin{bmatrix} 2 & 1 \end{bmatrix} \times \begin{bmatrix} 3 \\ 5 \end{bmatrix}$

Solution

$(2)(3) + (1)(5) = 6 + 5 = 11$ (result is a $1 \times 1$ matrix, or just the number 11)

Key Takeaways

Addition/subtraction: same-size matrices, add corresponding entries
Scalar multiplication: multiply every entry by the number
Matrix multiplication: row × column, then sum (columns of A must equal rows of B)
Determinant of $2 \times 2$ : $ad - bc$ (cross-multiply and subtract)
Matrix questions on the ACT are usually straightforward — don't skip them!

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ACT Math Matrices: Basic Operations and Applications

In this article

What Is a Matrix?

Matrix Addition and Subtraction

Scalar Multiplication

Matrix Multiplication

The Identity Matrix

Determinant of a 2×2 Matrix

Real-World Applications

Practice Problems

Key Takeaways

Written by Julien

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