Adding and multiplying matrices · Matrices & determinants

A matrix is a rectangle of numbers in brackets. Its order is (number of rows) × (number of columns): a matrix with 2 rows and 3 columns is a $2 \times 3$ matrix. Two matrices are equal only if they have the same order and every entry matches. So equal matrices give an equation for each entry.

OrderRows first, then columns: this is 2 × 3

Equal matricesSame place, same value: one equation per entry

Adding, subtracting and multiplying by a number

Matrices of the same order are added or subtracted entry by entry. To multiply a matrix by a number, multiply every entry by it.

AddingAdd the entries in the same place

Times a numberMultiply every entry by k

\begin{aligned} &3\begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix} - \begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix} \\ &= \begin{pmatrix} 3 - 2 & 6 - 5 \\ 0 - 1 & -3 - 3 \end{pmatrix} \\ &= \begin{pmatrix} 1 & 1 \\ -1 & -6 \end{pmatrix} \end{aligned}

Multiplying two matrices

To multiply two matrices, each entry of the answer comes from a row of the first matrix and a column of the second. Multiply them in pairs and add. For example, the entry in row 1, column 2 uses row 1 of the first and column 2 of the second.

This only works when the first matrix has as many columns as the second has rows. A $2 \times 2$ matrix times a $2 \times 1$ matrix gives a $2 \times 1$ answer.

Row times columnRow 1 of the first times column 1 of the second gives the top-left entry

Try it

Multiplying matricesTap an entry of the answer

2135

1420

2, 1row 1 of the first1, 2column 1 of the second2 × 1 + 1 × 2 = 4entry in row 1, column 1

The entry in row 1, column 1 uses row 1 of the first matrix and column 1 of the second: multiply them in pairs and add, 2 × 1 + 1 × 2 = 4. The first matrix needs as many columns as the second has rows.

Tap each entry of the answer and watch which row and column light up. Then switch on “Your turn” and work the entries out yourself.

The identity matrix and powers

The identity matrix $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ acts like the number 1: $AI = IA = A$ . A power such as $P^2$ means $P \times P$ , worked out row times column. It is not each entry squared.

The identity1s down the leading diagonal, 0s elsewhere

A powerP² = P × P, row by column

A number times $I$ puts that number on the leading diagonal only: $4I = \begin{pmatrix} 4 & 0 \\ 0 & 4 \end{pmatrix}$ , not a matrix of 4s. So $P^2 - 2P + 4I$ adds 4 to the two diagonal entries and nothing else.

Worked example · JAMB 2001

JAMB 2001 · UME · Q18

Given the matrix $K = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ , the matrix $K^2 + K + I$ , where $I$ is the $2 \times 2$ identity matrix, is

K squared
- ${K^2 = \begin{pmatrix} 4 + 3 & 2 + 4 \\ 6 + 12 & 3 + 16 \end{pmatrix}}$ .
- So ${K^2 = \begin{pmatrix} 7 & 6 \\ 18 & 19 \end{pmatrix}}$ .
Think first. K × K, row by column.
Add K and I, place by place
- Top row: ${7 + 2 + 1 = 10}$ and ${6 + 1 + 0 = 7}$ .
- Bottom row: ${18 + 3 + 0 = 21}$ and ${19 + 4 + 1 = 24}$ .
- So ${K^2 + K + I = \begin{pmatrix} 10 & 7 \\ 21 & 24 \end{pmatrix}}$ : option B.
Think first. I adds 1 on the leading diagonal only.

Finding unknown entries

When a product with unknowns equals a given matrix, work out the product with the unknowns in it. Then match entries: each entry gives an equation.

Worked example · WAEC 2017

WAEC 2017 · Paper 2 · Q12 (a)

If $\begin{pmatrix} 3x & 2 \\ 5 & 4 \end{pmatrix}\begin{pmatrix} 3 & -2 \\ 5 & 2y \end{pmatrix} = \begin{pmatrix} 28 & 4 \\ 35 & 22 \end{pmatrix}$ , find the values of $x$ and $y$ .

Multiply out
- Top row: ${3x(3) + 2(5) = 9x + 10}$ and ${3x(-2) + 2(2y) = -6x + 4y}$ .
- Bottom row: ${5(3) + 4(5) = 35}$ and ${5(-2) + 4(2y) = -10 + 8y}$ .
Think first. Work out each entry of the product, row by column, keeping x and y as letters.
Match the top-left entry
${9x + 10 = 28}$ , so ${9x = 18}$ and ${x = 2}$ .
Think first. Which entry has only x in it?
Match the bottom-right entry
${-10 + 8y = 22}$ , so ${8y = 32}$ and ${y = 4}$ .
Think first. Which entry has only y in it?
Check with another entry
$-6(2) + 4(4) = -12 + 16 = 4$ ✓.
Think first. Does the top-right entry agree?

Your turn

WAEC 2017 · Paper 2 · Q12 (b)

(b)
Given that $M = \begin{pmatrix} 1 & 2 \\ 4 & 3 \end{pmatrix}$ , $N = \begin{pmatrix} m & x \\ n & y \end{pmatrix}$ and $MN = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ , find the matrix $N$ .
Show the answer
$N = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

Worked solution (try it first)

(b)

Multiply row by column: $MN = \begin{pmatrix} m + 2n & x + 2y \\ 4m + 3n & 4x + 3y \end{pmatrix}$ .
Match each entry with $\begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ .
The first column gives $m + 2n = 2$ and $4m + 3n = 3$ .
Take 4 times the first from the second: $-5n = -5$ , so $n = 1$ and then $m = 0$ .
The second column gives $x + 2y = 1$ and $4x + 3y = 4$ .
In the same way, $-5y = 0$ , so $y = 0$ and then $x = 1$ .
So $N = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ .
Check: $\begin{pmatrix} 1 & 2 \\ 4 & 3 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ ✓.

Report a problem with this question