## Saturday, June 7, 2008

### Ch. 11 Matrices - Inverse - Concept Review 3

Inverse of a matrix
Let A be a square matrix of order n

If AB = In = BA

The B is inverse of A and is written as
A-1 = B

Theorems related to Inverses of matrices

1. Every invertible matrix possesses a unique inverse

2. A square matrix is invertible iff it is nonsingular.

4. Cancellation laws: Let A, B, and C be square matrices of the same order n. If A is a non-singular matrix, then

(i) AB = AC => B = C … (left cancellation law)
(ii) BA = CA => B = C … (right cancellation law)

This law is true only when |A| ≠ 0. Otherwise, there may be matrices such that AB = AC but B≠C.

5. Reversal law: If A and B are invertible matrices of the same order, then AB is invertible and

(AB) -1 = B-1A-1

6.If A,B,C are invertible matrices then
(ABC) -1 = C-1B-1A-1

7.If A is an invertible square matrix, then AT is also invertible and
(AT)-1 = (A-1)T

8. Let A be a non-singular square matrix of order n. Then

9. If A and B are non-singular square matrices of the same order, then