Lesson Objective

Dear learner,

by the end of this lesson, you will be able to:

• Determine the determinant of a square matrix of order 2
• Find the minor and cofactor of a given element of a matrix
• Calculate the determinant of a square matrix of order 3

Key Terms

• Minor
• Cofactor
• Determinant

Brainstorming Activity

Dear learner, how was the previous lesson! I think it was nice now consider $A = \begin{pmatrix}a & b \\c & d \end{pmatrix}$ be a 2 X 2 matrix then observe what happens when you delete

i. First row and first column

ii. First row and second column

iii. Second row and first column

iv. Second row and second column

Solution:

i. deleting first row and column leads obtaining entry “d” which is determined to be minor of $a_{1,1}$

ii. deleting first row and second column leads obtaining entry “c” which is determined to be minor of $a_{12}$

iii. deleting second row and first column leads obtaining entry “b” which is determined to be minor of $a_{21}$

iv. deleting second row and second column leads obtaining entry “a” which is determined to be minor of $a_{22}$

1.1. Determinant of Asquare matrix

Definition : Let A be square matrix with order n. then determinant of A is a scalar value of  a function of entries of A. The determinant of A is denoted by: $det(A)$ or $|A| \in \mathbb{R}$

Example 1:

1.Let A =$\begin{pmatrix}a&b&c&\\e&f&g\\p&q&r\end{pmatrix}_{3\times 3}$, then determinant of A = det(A) =$\begin{vmatrix}a&b&c&\\e&f&g\\p&q&r\end{vmatrix}$.

i. Determinants of Matrices with Order 2

Definition:

Let $A = \begin{pmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{pmatrix}$ be as quare matrix of order2 or (a 2 X 2 matrix),then determinant of A is defined as: $det(A) = |A| = a_{11} a_{22} – a_{12}a_{21}$

Example 2:

1. lLet A = $\begin{pmatrix}4&5\\7&2\end{pmatrix}$, then find

a). det(A)

b). det(2A)

2.Let A = $\begin{pmatrix}x&3+x\\1&2x\end{pmatrix}$ and |A| = 3, then find x?

Solution:

1a). det(A) = $\begin{vmatrix}4&5\\7&2\end{vmatrix}$ =$(4\times2) – (5\times 7)$ = 8 – 35 = -27

b). det(2A) = det(2$\begin{pmatrix}4&5\\7&2\end{pmatrix})$ =$\begin{vmatrix}2\times 4&2\times5\\2\times7&2\times2\end{vmatrix}$ = $8\times4 – 10\times14$ = -108

2.det(A) = 3 = $\begin{vmatrix}x&3+x\\1&2x\end{vmatrix}$

$\implies$ $2x^{2} – x – 3 = 3$

$\implies$ $2x^{2} – x – 6 = 0$ if and only if $2x(x -2) +3(x -2) = 0$

$\implies$ $(2x+3)(x -2) = 0$

$\implies$ $x = -\frac{3}{2}$ or $x = 2$.

1.2. Minors and Cofactors of Elements of Matrices

Definition: Let $A = (a_{ij})_{n}$ be a matrix , where $n \in \mathbb{Z^+}$ and M be the sub matrix obtained by deleating $i^{th}$rows and $j^{th}$ column of A.

then: i. The minor of A at (i,j) is $M_{ij}(A)$ is the determinant of sub matrix $A_{ij}$. i.e. $M_{ij} = det(A_{ij})$

ii the cofactor $C_{ij}(A)$ is the signed determinant of the sub matrix $A_{ij}$. i.e $C_{ij} = (-1)^{i+j}M_{ij}$.

The cofactor $C_{ij}(A)$ at location (i,j) can be computed as:

$C_{ij}(A) = \begin{cases} det(M_{ij} & \text{if } i + j = even \\ -det(M_{ij} & \text{if } i + j = odd\end{cases}$

Example 3:

1.Let A = $\begin{pmatrix}1&2&3\\5&2&1\\0&3&2\end{pmatrix}$ then find the minors and cofactors of A?

Solution:

NOTE:

The sign $(-1)^{i+j}$ on the cofactor of a matrix is a pattern with +’s on the main diagonal.

$\begin{pmatrix}+&-&+\\ -&+&-\\+&-&+\end{pmatrix}$

1.3. Determinant of A Matrix with Order 3

Definition:

Let A = $\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}$ be a $3\times3$ matrix, then:

det(A) =$\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}$ such that

det(A) =$a_{11}C_[{11} + a_{12}C_{12} +a_{13}C_{13}$ ………… first row expansion

det(A) = $a_{21}C_{21} + a_{22}C_{22} +a_{23}C_{23}$ ……… second row expansion

det(A) = $a_{31}C_{31} + a_{32}C_{32} +a_{33}C_{33}$ …….. third row expansion and it is said to be the cofactor expansion.

https://www.geogebra.org/m/ku5sugtc

Example 4:

1.Find the determinant of the following matrices

a). A = $\begin{pmatrix}1&2&3\\3&1&0\\2&3&5\end{pmatrix}$

b). B = $\begin{pmatrix}2&3&0\\1&1&2\\0&2&0\end{pmatrix}$

Solution:

a). det(A) =$\begin{vmatrix}1&2&3\\3&1&0\\2&3&5\end{vmatrix}$ = $1\times \begin{vmatrix}1&0\\3&5\end{vmatrix} +(-2)\begin{vmatrix}3&0\\2&5\end{vmatrix} + 3\begin{vmatrix}3&1\\2&3\end{vmatrix}$ = (5 – 0) +(-2)(15 – 0) + 3(9 – 2) = 5 – 30 + 21 = -13.

det(A) = -13

b). det(B) =$\begin{vmatrix}2&3&0\\1&1&2\\0&2&0\end{vmatrix}$ = $2\times \begin{vmatrix}1&2\\2&0\end{vmatrix} +(-3)\begin{vmatrix}1&2\\0&0\end{vmatrix} + 0\begin{vmatrix}1&1\\0&0\end{vmatrix}$ = 2(0 – 4) -3(0) + 0(2 – 0) = -8 – 0 + 0 = -8.

Note:

Determinant of an $n\times n$ matrix$A =(a_{ij})_{n\times n}$ is a real number obtained by:

$det(A) = a_{i1}C_{i1} +a_{i2}C_{i2} + … +a_{in}C_{in}$ …….. $i^{th}$ row expansion

$det(A) = \sum_{j=1}^{n}a_{kj}C_{kj}$, k = 1, 2, 3, …, n. ……….. $k^{th}$ row expansion