Matrix multiplication

Introduction to matrices and multiplication

A matrix (plural: matrices) is a rectangular array of numbers arranged in rows and columns. Matrices are commonly used to organize information and perform structured calculations. On the ACT, matrices are tested as a procedural skill—you are expected to apply rules accurately rather than understand advanced theory.

A matrix is written using brackets, like this:

$\left[\begin{array}{ccc}2& 5& -1\\ 4& 0& 3\end{array}\right]$

This matrix has:

• 2 rows
• 3 columns

We describe its size (also called its dimensions) as:

$$2\times 3$$

Rows come first, columns second.

This order is extremely important when multiplying matrices.

Scalar multiplication of a matrix

A scalar is simply a regular number. Scalar multiplication means:

Multiply every entry in the matrix by the scalar.

Example: Scalar multiplication

Multiply the matrix by $-2$:

$-2\left[\begin{array}{ccc}3& -1& 4\\ 0& 5& 2\end{array}\right]$

Result:

$\left[\begin{array}{ccc}-6& 2& -8\\ 0& -10& -4\end{array}\right]$

Multiplying two matrices

Matrix multiplication is more structured and requires careful attention to rows, columns, and order.

When can two matrices be multiplied?

Two matrices can be multiplied only if:

The number of columns in the first matrix equals the number of rows in the second matrix.

Visual dimension rule

$$\boxed{2\times 3}\cdot \boxed{3\times 2}\text{✓ Allowed}$$

$$\boxed{2\times 3}\cdot \boxed{4\times 2}\text{✗ Not Allowed}$$

Size of the resulting matrix

If:

$$(2\times 3)\cdot (3\times 2)$$

Then the product matrix will be:

$$2\times 2$$

The outside numbers determine the size of the result.

How to do matrix multiplication

To find each entry in the product matrix, take one row from the first matrix and one column from the second matrix. Multiply the corresponding numbers and add the results.

• The top-left entry comes from the first row of the first matrix and the first column of the second matrix.
• The top-right entry comes from the first row and the second column.
• This process repeats for each row.

Example: Multiplying two matrices

Multiply the matrices below.

$A=\left[\begin{array}{ccc}2& -1& 4\\ 0& 3& -2\end{array}\right](2\times 3)$

$B=\left[\begin{array}{cc}5& -2\\ 1& 4\\ -3& 0\end{array}\right](3\times 2)$

Step 1: Check dimensions

$(2\times 3)\cdot (3\times 2)\text{✓ Allowed}$
The result will be a $2\times 2$ matrix.

Step 2: Compute each entry

Top-left entry (row 1 $\times$ column 1)
$(2\cdot 5)+(-1\cdot 1)+(4\cdot -3)=10-1-12=-3$

Top-right entry (row 1 $\times$ column 2)
$(2\cdot -2)+(-1\cdot 4)+(4\cdot 0)=-4-4+0=-8$

Bottom-left entry (row 2 $\times$ column 1)
$(0\cdot 5)+(3\cdot 1)+(-2\cdot -3)=0+3+6=9$

Bottom-right entry (row 2 $\times$ column 2)
$(0\cdot -2)+(3\cdot 4)+(-2\cdot 0)=0+12+0=12$

Final product matrix

$AB=\left[\begin{array}{cc}-3& -8\\ 9& 12\end{array}\right]$

Order matters: $AB\ne BA$

Matrix multiplication is not commutative.

Even if $AB$ in this example is defined, $BA$ may be **undefined **if the inner dimensions do not match. Even if they do match, the resulting matrix will have different values or different dimensions than $AB$.

Switching the order can completely change—or eliminate—the result.