MATRICES II
Mathematics SSS 3 First Term
WEEK 2
MATRICES II
Performance Objectives
Student should be able to:
- Perform matrix multiplication of 2
2 and 3
3 matrices. - Find the transpose of a matrix.
- Calculate the determinant of matrices.
- Solution to simultaneous equation using determinant method.
- Inverse of 2
2 matrix.
Content
Let 
and 
The product of the matrices 
and 
wriiten as 
or 
is a matrix 
such that
Therefore, two matrices 
and 
can be multiplied together only when the number of columns of the first matrix 
is equal to the number of rows of the second matrix 
. Now suppose 
is a matrix of order 
and 
is a matrix of order 
, the product of 
and 
, written 
can be obtained since the number of columns of 
i.e. 2 is equal to the number of rows B
i.e. 2 and this product 
is a matrix of order
.
Example 1: Let A= 230-245
and B=132-106
. Find
- AB
ii. BA
Solution:
- AB
is defined since the number of colums of A
is equal to the number of row of B
and AB
is a 3 ×
3 matrix therefore
- Since
is a (2 
3) matrix and 
is a (3 
2) then 
is defined and is a (2
2) matrix. Therefore
Note that for two matrices 
and 
Example 2: Given that
and 
. Compute the following:
ii. 
iii. 
Solution:
- Since
is a (2 
2) and 
is (2 
3) then 
is defined and it is a (2 
3) matrix. therefore
- Since
is a (2 
3) and 
is (2 
2), the number of columns of 
(i.e. 3) is not equal to the number of rows of 
(i.e. 2, hence (
) is not defined and thus cannot be computed.
, solve the bracket first
is defined since 
is a (2 
2) matrix and 
is of order 
Using 
and 
as stated in example 2(iii) above show that 
Transpose of a Matrix
The transpose of a matrix A
written as AT
is a matrix gotten by interchanging the row and columns of A
i.e. the first row become the first column while the second row become the second column etc.
Example 3: if 
and 
Find
ii. 
iii. 
iv. 
Solution
is defined since 
is (2 
3) and 
is (3 
2)
- From i. and ii.
is defined since 
is of order (2 
3) and 
is of order (3 
2)
Ask students to say what they notice about the result in iii. And iv.
Determinant of (2 ×
2) and (3 ×
3)Matrices
Determinant of (2 ×
2)
Let 
be a 2 by 2 matrix
then the determinant of 
denoted by 
or det 
is given by
Determinant of (3 ×
3)
Let 
be a 3 by 3 matrix, then
Where
are called minors and they are obtained by deleting respective leading row and column- The sign attached to b11, b12, b13
on and on is the cofactor signs given by the chess board format 
or 
where 
is row and 
-
and so on
Example 4: If 
and 
Find
- det
ii. det 
iii. 
Solution
- ∴det
- ∴det
- From i. and ii. above
Example 5: If 
, Find N
Solution
Example 6: Find the determinant of the following matrix in terms of x
Solution
Inverse of (2 ×
2) Matrix
If 
then the inverse denoted as 
is given as exchanging 
and 
and negating 
and 
Example 7: Find the inverse of matrix 
Where 
Given that 
show that 
Solution of Simultaneous Equation Using Determinant Method
(Crammer’s Rule)
Two Equation in Two Unknown
Given the equations
Where 
are all constants. To solve for 
and 
, we put the coefficient in determinant form as follows:
Find the determinant 
, then
is obtained by replacing the column of 
by 
.
is obtained by replacing the column of 
by 
then divide 
by
to get 
and 
by 
to get 
. i.e.
Example 8: use the determinant method to solve the following simultaneous equation
Solution
Putting the coefficients in determinant form, we have
Therefore x=2
and y=-1
Three Equation in Three Unknown
Given the equations
Where 
are all constants. To solve for x, y
and z
, we put the coefficient in determinant form as follows:
is obtained by replacing
by 
is obtained by replacing 
by
is obtained by replacing 
by 
Therefore
Example 9: Solve the following simultaneous equations using the determinant method
Solution
Putting the coefficient in the determinant form we have
Therefore 
and 