Matrices

A Grid of Numbers With Rules

A matrix is a rectangular array of numbers arranged in rows and columns. We describe its size as rows $\times$ columns, so a matrix with 2 rows and 3 columns is a "$2 \times 3$ matrix." The ACT does not ask you to prove anything deep about matrices — it asks whether you can add, subtract, scale, multiply, and take the determinant of small matrices without slipping on the mechanics. Because these operations are procedural, matrix questions are among the most learnable points on the entire test.

Adding and Subtracting

You can add or subtract two matrices only when they have the same dimensions. Combine them entry by entry — the number in row 1, column 1 of the first matrix pairs with the number in row 1, column 1 of the second, and so on.

$\begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix} + \begin{bmatrix} 4 & -1 \\ 0 & 6 \end{bmatrix} = \begin{bmatrix} 6 & 4 \\ 1 & 9 \end{bmatrix}$

Subtraction works the same way; just be careful with signs when a negative entry meets a subtraction.

Scalar Multiplication

Multiplying a matrix by a single number (a scalar) multiplies every entry by that number:

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

Do not multiply just the first entry or just the first row — the scalar reaches every cell.

Matrix Multiplication

Multiplying two matrices is the operation students most often get wrong, because it is not entry-by-entry. To multiply matrix $A$ by matrix $B$, the number of columns of $A$ must equal the number of rows of $B$. Each entry of the product is a row-times-column dot product: to get the entry in row $i$, column $j$, march across row $i$ of $A$ and down column $j$ of $B$, multiply matching numbers, and add.

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1\cdot5 + 2\cdot7 & 1\cdot6 + 2\cdot8 \\ 3\cdot5 + 4\cdot7 & 3\cdot6 + 4\cdot8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$

The product of an $m \times n$ matrix and an $n \times p$ matrix is an $m \times p$ matrix — the inner dimensions must match and they "cancel," leaving the outer dimensions as the size of the answer. Matrix multiplication is not commutative: $AB$ and $BA$ are usually different, and one may not even be defined.

The Determinant of a 2×2 Matrix

For a square $2 \times 2$ matrix, the determinant is a single number that captures how the matrix scales area (and whether it is invertible):

$\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$

It is the product of the main diagonal minus the product of the off-diagonal. For $\begin{bmatrix} 3 & 5 \\ 2 & 4 \end{bmatrix}$ the determinant is $(3)(4) - (5)(2) = 12 - 10 = 2$. A determinant of $0$ signals a matrix with no inverse.

Dimension Bookkeeping

Operation	Requirement	Result size
$A + B$, $A - B$	Same dimensions	Same as $A$
$kA$ (scalar)	Any matrix	Same as $A$
$A B$ (product)	cols of $A$ = rows of $B$	rows of $A$ $\times$ cols of $B$
$\det A$	Square matrix	A single number

A Reliable Game Plan

• Check dimensions first. For adding/subtracting they must match; for multiplying, the inner dimensions must match.
• Add and scale entry by entry; the scalar or the sum touches every cell.
• Multiply by dot-producting rows into columns, never cell by cell.
• Determinant of a $2\times2$ is main diagonal minus off-diagonal: $ad - bc$.
• Remember order matters in multiplication — keep $A$ on the left when the problem writes $AB$.

Because every step is mechanical, the way to earn these points is to slow down, label the dimensions, and execute the arithmetic cleanly.