Find the inverse of the following matrices by gauss eliminat

Find the inverse of the following matrices by gauss elimination: B = (1 2 5 0 1 4 0 0 1) C = (-4 0 0 0 8 3 0 13 5) Double-check your result by multiplying the matrix with the inverse you found.

Solution

1

0

0

1

0

0

2

1

0

0

1

0

5

4

1

0

0

1

We will reduce A to itsREF as under:

Add -2 times the 1st row to the 2nd row

Add -5 times the 1st row to the 3rd row

Add -4 times the 2nd row to the 3rd row

Then the RREF of A is

1

0

0

1

0

0

0

1

0

-2

1

0

0

0

1

3

-4

1

Therefore, B^-1 =              

1

0

0

-2

1

0

3

-4

1

On verification, we find that BB^-1 = B^-1B = I₃

2. Let A =                 

-4

0

0

0

8

13

0

8

13

0

1

0

0

3

5

0

0

1

We will reduce A to itsREF as under:

Multiply the 1st row by -1/4

Multiply the 2nd row by 1/8

Add -3 times the 2nd row to the 3rd row

Multiply the 3rd row by 8

Add -13/8 times the 3rd row to the 2nd row

Then the RREF of A is

1

0

0

-1/4

0

0

0

1

0

0

5

-13

0

0

1

0

-3

8

Therefore, C^-1 =

-1/4

0

0

0

5

-13

0

-3

8

On verification, we find that AA^-1 = A^-1A = I₃

                               

1	0	0	1	0	0
2	1	0	0	1	0
5	4	1	0	0	1