Page 30 - mathematics

Determinant of a matrix Example

(1) Determinant of 2  2 Matrix): Find A of the following matrices:

If A  a11 a21  i  A  4 3 1 2 2
a12 a22   2 1 
ii  A  2 5 1
Then det A  A  a11a22  a21a12
4 5 3

(2) Determinant of 3  3 Matrix): Solution i  A  4 3  (4)(1)  (3)(2)  2
2 1
a11 a21 a31 
If A  a12 
a22 a32  1 2 2 5 1 2 1 2 5
5 1 5 3 4 3 4 5
a13 a23 a33  ii  A  2 5 3  (1)  (2)  (2)

Then A  a11 a22 a23  a21 a21 a23  a13 a21 a22 4
a32 a33 a31 a33 a31 a32
 10  (2) 2  (2) 10  14

Inverse of a matrix A 1 Example

a11 a12  a1n  Find A 1 of the following matrix:
a21  1 2 2
If A   a22  a2n  ,
   A  2 5 1
 4 5 3

an1 an 2  ann  Solution

1 2 2
Then A 1  1 adj A  A  2 5 1  14
A
4 5 3
where:

adj A   adj aij   Aij T

Aij is the cofactor of each element.

 10  2  10 T Solution of linear system of equations
  5  by using Inverse matrix method
adj  A     4   3  3 
* We will solve Linear system of equations which take
  8  1  the following form:

10 4 8  a11x 1  a12x 2   a1n x n  b1
a21x 1  a22x 2   a2n x n  b2
  2 5 3 

10 3 1  an1x 1  an2x 2   ann x n  bn
to get the values of the unknowns x 1, x 2,, x n .
1 10 4 8
14 5 3
A 1   2 3 1 

10

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