MATRIX DETERMINANT

In this page matrix determinant we are going to see how to find determinant for any matrix and examples based on this topic.

Definition :

For every square matrix A of order n with entries as real or complex numbers, we can associate a number called determinant of matrix A 

It is denoted by |A| or det (A) or ∆.

Example 1 :

Find determinant of the following matrix.

 
3 4 1
0 -1 2
5 -2 6
 

Solution :

A=
 
3 4 1
0 -1 2
5 -2 6
 

   = 3

 
-1 2

-2 6
 

-4

 
0 2

5 6
 

+1 

 
0 -1

5 -2
 

    =  3 [ -6-(-4) ] -4 [ 0-10 ]+1 [0-(-5)]

    =  3 [ -6+4 ] -4 [ -10 ]+1 [5]

    =  3 [ -2 ] -4 [ -10 ]+ 1 [5]

    =  -6 + 40 + 5

    =  -6 + 45

    =  39 

Example 2 :

Find determinant of the following matrix.

 
1 1 -1
2 1 -2
1 -1 1
 

Solution :

= 1

 
1 -2

-1 1
 

 -1

 
2 -2

1 1
 

 -1

 
2 1

1 -1
 

    =  1 [1-2 ] -1 [ 2-(-2) ] - 1 [-2-1]

    =  1 [-1] -1 [ 2+2 ] - 1 [-3]

    =  -1 -1 (4) + 3

    =  -1 -4 + 3

    =  -5 + 3

    =  -2

Example 3 :

Find determinant of the following matrix.

 
1 2 3
-1 3 4
2 0 -1
 

Solution :

=1

 
3 4

0 -1
 

-2

 
-1 4

2 -1
 

+3

 
-1 3

2 0
 

    =  1 [ -3 - 0 ] -2 [ 1 - 8 ] + 3 [0 - 6]

    =  1 [ -3 ] -2 [ -7 ] + 3 [- 6]

    =   -3 + 14 - 18

    =   -21 + 14

    =   -7               matrix determinant   matrix determinant  matrix determinant

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