Matrix Operations

A matrix is a rectangular array of numbers arranged in rows and columns. Matrix operations are used in solving systems of equations, transformations, and more. You’ll need to know how to add, subtract, and multiply matrices, as well as find their scalar multiples.

 

Matrix Notation

A matrix is usually written in square brackets:

$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$

• This is a 2×2 matrix (2 rows, 2 columns)
• The number in the first row and second column is 2

 

Matrix Addition and Subtraction

Add or subtract corresponding elements — this means element by element.

Example:

Let

$$A = \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 1 \\ 7 & 2 \end{bmatrix}$$

Then:

$$A + B = \begin{bmatrix} 2 + 4 & 5 + 1 \\ 1 + 7 & 3 + 2 \end{bmatrix} = \begin{bmatrix} 6 & 6 \\ 8 & 5 \end{bmatrix}$$

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

 

Scalar Multiplication

To multiply a matrix by a number (called a scalar), multiply each element in the matrix by that number.

Example:

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

 

Matrix Multiplication (2 × 2)

To multiply two matrices $A \times B$:

• Multiply the rows of A by the columns of B.
• This operation is not element-by-element!

Important: Matrix multiplication is only defined when the number of columns in the first matrix equals the number of rows in the second.

Example:

Let

$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 0 \\ 1 & 5 \end{bmatrix}$$

$$AB = \begin{bmatrix} (1\times2 + 2\times1) & (1\times0 + 2\times5) \\ (3\times2 + 4\times1) & (3\times0 + 4\times5) \end{bmatrix} = \begin{bmatrix} 4 & 10 \\ 10 & 20 \end{bmatrix}$$

 

Zero and Identity Matrices

• Zero matrix: All elements are 0
• Identity matrix: 1s on the diagonal, 0s elsewhere

$$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad A \times I = A$$

 

Worked Example