# #FocusOnExams Mathematics GCE ‘A’ L – 28th April 2020

Tuesday, 28th April 2020

TOPIC: MATRICES

Lesson: Determinant of a 3 by 3 matrix

Objectives:

By the end of this lesson, learners should be able to:

• Find the determinant of 3 by 3 matrices

Review of last lesson:

Definition of a matrix:

This is a rectangular array of numbers, symbols, or expressions arranged in rows and columns

Matrix notation:

Matrices are denoted with upper case (capital) letters, and their elements with lower case (small) letters with subscripts to represent individual entries or elements in the matrices.

Order or size or dimension of a matrix:

This is given by the number of rows and columns in that order, which formed the matrix.

Addition or subtraction of any two matrices is only possible, if they have the same order, size or dimension. Addition or subtraction of matrices is done by adding or subtracting corresponding elements

Multiplication of a matrix by a scalar:

This is done by multiplying each element of the matrix, by the scalar, where the scalar is any real number.

Equality of matrices:

Two matrices are said to be equal, if they have the same order, size or dimension and  their corresponding elements are all equal.

Multiplication of matrices:

To multiply a matrix by another matrix,

• The number of columns in the first matrix must be the same as the number of rows in the second matrix (known as the pre-requisite).
• Multiply the elements of each row of the first matrix by the elements of each column in the second matrix, and then add the products.

Determinant of a 3 by 3 matrix:

Given a matrix, , where .

The determinant of matrix , is denoted as:

– det( or

– .

Example 1

Given the matrix, , where . Find the determinant of .

solution

Method 1

det() =

det( = [0 + 0 + 4] – [0 + (-6) + 0]

det() = 4 + 6

det( = 10

Method 2

We expand the 3 by 3 matrix along any row or column, making use of the sign scheme. Using the first row for convenient to find the det(, we have:

det( =

det( =

det( = 3(0 + 2) – 0 (2 – 0) + 2(2 – 0)

det( = 6 – 0 + 4

det( = 10.

Example 2

Find the determinant of the matrix, .

Solution

Method 1

Let

det( =

det( = [12 + (- 18) + (- 28)] – [12 + (-28) + (-18)]

det( = – 34 + 34

det( = 0

Method 2

Using the sign scheme , and expanding the matrix along the 2nd column, we have:

det( =
det( =

det( =

det( = 0.

You can use any of the methods above to find the determinant of a 3 by 3 matrix.

Note

1. In general, if is a matrix, and  where ,

then

, where .

1. If and are any two matrices and .

,

then

, where .

Example 3

is a  matrix such that, . Find, .

Solution

is a  matrix and

=

= .

Example 4

If  and  are any two matrices, and  and , find .

Solution

and

=

=

Assignment

Exercise 17 b, page 597 of Explaining pure mathematics for Advanced level by Napthalin Atanga

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