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.
3. A-1 = (1/|A|)adj A
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
|adj A| = |A|n-1
9. If A and B are non-singular square matrices of the same order, then
adj AB = (adj B) (adj A)
10. If A is an invertible square matrix, then
adj AT = (adj A) T
11. If A is a non-singular square matrix, then
adj(adj A) = |A|n-2A
No comments:
Post a Comment