Inverse of a matrix

Let A be a square matrix of order n

If AB = I_n = 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 A^T is also invertible and
(A^T)^-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 A^T = (adj A) ^T


11. If A is a non-singular square matrix, then

adj(adj A) = |A|^n-2A