a c Write the matrix A as a product of elementary matrices A

(a) c Write the matrix A as a product of elementary matrices: A = [1 -2 0 0 1 -1 0 0 1] (b) Find the inverses of each of the elementary matrices you found in part. (c) Find the inverse of the matrix A, by using the elementary matrices from part.

Solution

We know that an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. In order to express A as a product of elementary matrices, we first, reduce A to its RREF as under:

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

Add 1 times the 2nd row to the 3rd row

Then, the RREF of A = I₃.

Now, let E₁ =

1

0

0

2

1

0

0

0

1

and E₂ =

1

0

0

0

1

0

0

1

1

Then E₂ E₁ A = I₃

Therefore, A = E₁^-1 E₂^-1 ( the inverse of an elementary matrix is also an elementary matrix).

(b) We have E^-1 =

1

0

0

-2

1

0

0

0

1

and E₂^-1 =

1

0

0

0

1

0

0

-1

1

(c) Since_, A = E₁^-1 E₂^-1 we have A^-1 = (E₁^-1 E₂^-1 )^-1 = E₂ E₁ =

1

0

0

2

1

0

2

1

1

Note: (AB)^-1= B^-1 A^-1

1	0	0
2	1	0
0	0	1