Matrices and Determinants of Order 3

Lessons

Senior 5 · Mathematics

Matrices and Determinants of Order 3

5 lessons

Lesson 1 of 5

Square matrices of order 3 and their operations

35 min

Nine numbers in one object

A farming cooperative sells beans, maize and Irish potatoes at three markets: Musanze, Huye and Rubavu. Its weekly sales, in bags, fill a table with three rows (the markets) and three columns (the crops). Written as a single object, the table is a matrix, and adding two weeks, or comparing them, becomes one operation instead of nine. In S4 you worked with $2 \times 2$ matrices. This unit moves up to order 3, where matrices describe transformations of space, have determinants that measure volume, and solve systems of three equations.

Definition

A square matrix of order 3 is an array of 9 real numbers in 3 rows and 3 columns:

A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} = (a_{ij}) .

The entry $a_{ij}$ is in row $i$ and column $j$ . The entries $a_{11}, a_{22}, a_{33}$ form the leading (main) diagonal, and their sum is the trace: $\operatorname{tr} A = a_{11} + a_{22} + a_{33}$ . The set of all such matrices is written $M_3(\mathbb{R})$ .

Example: in $\begin{pmatrix} 2 & -1 & 0 \\ 4 & 3 & 7 \\ 5 & 1 & -6 \end{pmatrix}$ , $a_{23} = 7$ , $a_{31} = 5$ and the trace is $2 + 3 - 6 = - 1$ .

Types of square matrices

Type	Condition on the entries	Example
Zero matrix $O$	every $a_{ij} = 0$	$\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$
Diagonal	$a_{ij} = 0$ whenever $i \ne j$	$\begin{pmatrix} 3 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 5 \end{pmatrix}$
Scalar	diagonal, with all diagonal entries equal	$\begin{pmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{pmatrix}$
Identity $I_{3}$	scalar, with diagonal entries 1	$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$
Upper triangular	$a_{ij} = 0$ whenever $i > j$ (zeros below the diagonal)	$\begin{pmatrix} 2 & 5 & -1 \\ 0 & 3 & 4 \\ 0 & 0 & 6 \end{pmatrix}$
Lower triangular	$a_{ij} = 0$ whenever $i < j$ (zeros above the diagonal)	$\begin{pmatrix} 1 & 0 & 0 \\ 7 & 2 & 0 \\ -3 & 8 & 4 \end{pmatrix}$
Symmetric	$a_{ij} = a_{ji}$ for all $i, j$	$\begin{pmatrix} 1 & 2 & 3 \\ 2 & 0 & -4 \\ 3 & -4 & 5 \end{pmatrix}$
Skew-symmetric	$a_{ij} = -a_{ji}$ for all $i, j$	$\begin{pmatrix} 0 & 2 & -1 \\ -2 & 0 & 6 \\ 1 & -6 & 0 \end{pmatrix}$

Two matrices are equal when they have the same order and every pair of corresponding entries is equal. One matrix equation $A = B$ between $3 \times 3$ matrices is therefore nine equations between numbers.

Definition

Sum, difference and scalar multiple. For $A = (a_{ij})$ , $B = (b_{ij})$ of the same order and a real number $k$ :

A + B = (a_{ij} + b_{ij}), \qquad A - B = (a_{ij} - b_{ij}), \qquad kA = (k\,a_{ij}) .

Each operation works entry by entry. Example: $2\begin{pmatrix} 1 & 0 & 3 \\ -2 & 4 & 1 \\ 0 & 5 & -1 \end{pmatrix} = \begin{pmatrix} 2 & 0 & 6 \\ -4 & 8 & 2 \\ 0 & 10 & -2 \end{pmatrix}$ .

Important

Properties of addition and scalar multiplication. For $A, B, C \in M_3(\mathbb{R})$ and real numbers $h, k$ :

$A + B = B + A$ and $(A + B) + C = A + (B + C)$ ;
$A + O = A$ , and $A + (- A) = O$ where $- A = (- 1) A$ ;
$k (A + B) = k A + k B$ , $(h + k) A = h A + k A$ , $h (k A) = (h k) A$ and $1 A = A$ .

Each follows from the same property of real numbers, applied to every entry. Together they make $M_3(\mathbb{R})$ a vector space, the structure you meet again with $\mathbb{R}^3$ in the next unit.

Worked example 1: two weeks of sales

The cooperative's sales in bags (illustrative figures) are given by $A$ for week 1 and $B$ for week 2; rows are Musanze, Huye, Rubavu and columns are beans, maize, potatoes.

A = \begin{pmatrix} 12 & 20 & 35 \\ 8 & 15 & 40 \\ 10 & 18 & 25 \end{pmatrix}, \qquad B = \begin{pmatrix} 15 & 18 & 30 \\ 9 & 20 & 38 \\ 14 & 16 & 27 \end{pmatrix} .

Find the total sales over the two weeks and the change from week 1 to week 2. Which market sold fewer bags of potatoes in week 2?

Solution. Add and subtract entry by entry:

A + B = \begin{pmatrix} 27 & 38 & 65 \\ 17 & 35 & 78 \\ 24 & 34 & 52 \end{pmatrix}, \qquad B - A = \begin{pmatrix} 3 & -2 & -5 \\ 1 & 5 & -2 \\ 4 & -2 & 2 \end{pmatrix} .

The potato column of $B - A$ is $- 5, - 2, 2$ : Musanze sold 5 fewer bags and Huye 2 fewer, while Rubavu sold 2 more.

Worked example 2: a matrix equation

Solve $2 X - P = 3 Q$ for the matrix $X$ , where $P = \begin{pmatrix} 3 & -1 & 4 \\ 2 & 5 & 0 \\ -1 & 2 & 6 \end{pmatrix}$ and $Q = \begin{pmatrix} 1 & 3 & 0 \\ -2 & 1 & 4 \\ 3 & 0 & 2 \end{pmatrix}$ .

Solution. The rules above let you rearrange as with numbers: $2 X = P + 3 Q$ , so $X = \frac{1}{2}(P + 3Q)$ .

P + 3Q = \begin{pmatrix} 6 & 8 & 4 \\ -4 & 8 & 12 \\ 8 & 2 & 12 \end{pmatrix}, \qquad X = \begin{pmatrix} 3 & 4 & 2 \\ -2 & 4 & 6 \\ 4 & 1 & 6 \end{pmatrix} .

Check one entry: $2 (4) - (- 1) = 9 = 3 (3)$ . ✓

Definition

The transpose of $A = (a_{ij})$ is the matrix $A^T = (a_{ji})$ : row $i$ of $A$ becomes column $i$ of $A^{T}$ . Example:

\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}^T = \begin{pmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{pmatrix} .

With it, $A$ is symmetric when $A^{T} = A$ and skew-symmetric when $A^{T} = - A$ .

Important

Properties of the transpose: $(A^{T})^{T} = A$ , $(A + B)^{T} = A^{T} + B^{T}$ and $(k A)^{T} = k A^{T}$ .

Theorem. Every square matrix is the sum of a symmetric and a skew-symmetric matrix:

A = \underbrace{\tfrac{1}{2}(A + A^T)}_{S} + \underbrace{\tfrac{1}{2}(A - A^T)}_{K} .

Reason. $S^T = \frac{1}{2}(A^T + A) = S$ , so $S$ is symmetric; $K^T = \frac{1}{2}(A^T - A) = -K$ , so $K$ is skew-symmetric; and $S + K = A$ . Also, a skew-symmetric matrix has zeros on its diagonal, because $a_{ii} = -a_{ii}$ forces $a_{ii} = 0$ .

Worked example 3: unknown entries of a symmetric matrix

Find $a, b, c$ so that $M = \begin{pmatrix} 1 & a + 1 & 4 \\ 3 & 2 & b - c \\ 2b & 5 & 0 \end{pmatrix}$ is symmetric.

Solution. Symmetric means $m_{12} = m_{21}$ , $m_{13} = m_{31}$ , $m_{23} = m_{32}$ :

a + 1 = 3, \qquad 4 = 2b, \qquad b - c = 5 .

So $a = 2$ , $b = 2$ and $c = b - 5 = - 3$ .

Worked example 4: symmetric plus skew-symmetric

Write $N = \begin{pmatrix} 2 & 4 & -1 \\ 0 & 3 & 5 \\ 7 & 1 & 6 \end{pmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.

Solution. $N^T = \begin{pmatrix} 2 & 0 & 7 \\ 4 & 3 & 1 \\ -1 & 5 & 6 \end{pmatrix}$ , so

S = \tfrac{1}{2}(N + N^T) = \begin{pmatrix} 2 & 2 & 3 \\ 2 & 3 & 3 \\ 3 & 3 & 6 \end{pmatrix}, \qquad K = \tfrac{1}{2}(N - N^T) = \begin{pmatrix} 0 & 2 & -4 \\ -2 & 0 & 2 \\ 4 & -2 & 0 \end{pmatrix} .

Check: $S + K = N$ , $S$ is equal to its transpose, and $K$ has a zero diagonal with $k_{ij} = -k_{ji}$ . ✓

Warning

Only matrices of the same order can be added or subtracted. A $3 \times 3$ matrix plus a $3 \times 2$ matrix has no meaning.
The transpose reflects in the leading diagonal, not in the other diagonal: the diagonal entries stay where they are.
$k A$ multiplies every entry by $k$ , not just one row. (Multiplying one row by $k$ is a different operation; you will meet it with determinants.)

Tip

To spot a symmetric matrix, fold it along the leading diagonal in your mind: each entry must land on an equal entry. For a skew-symmetric matrix each entry lands on its negative, and the diagonal, which lands on itself, must be all zeros.

✦

Practice

Questions 1 to 3 are routine; 4 to 6 need more care; 7 asks for a proof.

For $A = \begin{pmatrix} 3 & -1 & 2 \\ 0 & 4 & 1 \\ 5 & 2 & -3 \end{pmatrix}$ , write $a_{23}$ , $a_{31}$ and $\operatorname{tr} A$ .
Name every type in the table that fits (a) $\begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix}$ , (b) $\begin{pmatrix} 1 & 4 & -2 \\ 0 & 3 & 7 \\ 0 & 0 & 5 \end{pmatrix}$ , (c) $\begin{pmatrix} 0 & 3 & -1 \\ -3 & 0 & 2 \\ 1 & -2 & 0 \end{pmatrix}$ .
With $A = \begin{pmatrix} 1 & 2 & 0 \\ -1 & 3 & 4 \\ 2 & 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 & -2 \\ 1 & 1 & 0 \\ 3 & -1 & 2 \end{pmatrix}$ , find $2 A - 3 B$ .
With the same $A$ and $B$ , solve $X + A^{T} = 2 B$ .
Find $x, y, z$ if $\begin{pmatrix} 0 & x - 2 & 3 \\ 5 & 0 & y \\ z + 1 & -4 & 0 \end{pmatrix}$ is skew-symmetric.
Write $M = \begin{pmatrix} 4 & 1 & 0 \\ 5 & -2 & 3 \\ 2 & 7 & 1 \end{pmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.
Prove that if $A$ and $B$ are symmetric, then $3 A - 2 B$ is symmetric. Is the same true for skew-symmetric matrices?

Show answersHide answers

$a_{23} = 1$ , $a_{31} = 5$ , $\operatorname{tr} A = 3 + 4 - 3 = 4$ .
(a) scalar, so also diagonal, upper and lower triangular, and symmetric. (b) upper triangular. (c) skew-symmetric.
$\begin{pmatrix} 2 & 1 & 6 \\ -5 & 3 & 8 \\ -5 & 3 & -4 \end{pmatrix}$
$X = 2B - A^T = \begin{pmatrix} -1 & 3 & -6 \\ 0 & -1 & 0 \\ 6 & -6 & 3 \end{pmatrix}$
$x - 2 = - 5$ , so $x = - 3$ ; $z + 1 = - 3$ , so $z = - 4$ ; $y = 4$ .
$S = \begin{pmatrix} 4 & 3 & 1 \\ 3 & -2 & 5 \\ 1 & 5 & 1 \end{pmatrix}$ , $K = \begin{pmatrix} 0 & -2 & -1 \\ 2 & 0 & -2 \\ 1 & 2 & 0 \end{pmatrix}$ .
$(3 A - 2 B)^{T} = 3 A^{T} - 2 B^{T} = 3 A - 2 B$ . Yes: if $A^{T} = - A$ and $B^{T} = - B$ , then $(3 A - 2 B)^{T} = - 3 A + 2 B = - (3 A - 2 B)$ .

Sources and credits

Written by Ganzaa: the explanations, the diagram, the worked examples, the practice items and the quiz questions. Every sum, difference and decomposition was checked by computer. The cooperative's sales figures are illustrative, not real data.

1Nine numbers in one object

2Types of square matrices

Nine numbers in one object

Types of square matrices