## 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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