Matrix Basics

A matrix is a rectangular array of numbers. An $m \times n$ matrix has $m$ rows and $n$ columns.

$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \quad \text{(2×2 matrix)}$

A 2×2 matrix with elements aᵢⱼ (row i, column j)

Addition/Subtraction: Add corresponding entries. Matrices must be the same size.

$A + B$
$[1,2; 3,4] + [5,6; 7,8]$ add corresponding elements
$[6,8; 10,12]$

Matrix addition: add corresponding elements

Scalar multiplication: Multiply every entry by the scalar.

Matrix multiplication: $A_{m \times n} \cdot B_{n \times p} = C_{m \times p}$ . The number of columns of $A$ must equal the number of rows of $B$ . Each entry: row of $A$ dot product with column of $B$ .

Example 1

Multiply $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 \\ 6 \end{pmatrix}$

Row 1: $1(5) + 2(6) = 5 + 12 = 17$

Row 2: $3(5) + 4(6) = 15 + 24 = 39$

Result: $\begin{pmatrix} 17 \\ 39 \end{pmatrix}$

\begin{pmatrix} 17 \\ 39 \end{pmatrix}

Tip

ACT Pro Tip

Matrix multiplication questions on the ACT are usually 2 $\times\$2 times 2$ \times$1 or 2 $\times\$2 times 2$ \times$2. The key: row times column, dot product style. Practice the pattern 3 times and it becomes automatic.

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