**Inverse of a Matrix**

If A is a
non-singular square matrix i.e. |A| ≠ 0, and there exists another square matrix
B such that AB = BA = I where I is an identity matrix of order same as that of
A or B, then the matrix B is said to be the **inverse of matrix** A and vice-versa. The inverse of A is denoted by
A^{-1}. ∴ B = A^{-1}.

__Note__**:** For the existence of the
inverse of a matrix, the following requirements are necessary:

i)
The matrix must be a square
matrix. This requirement is essential because, let A be a matrix of order 2×3
and B be another matrix of order 3×2, then AB and BA both exist but have
different orders, namely 2×2 and 3×3 and hence cannot be equal.

ii)
The equation AB = BA = I
(Identity Matrix) must be satisfied. For example, if

But, AB = BA ≠ I. So, they are not inverse matrix to each other.

iii) The matrix must be non-singular. i.e |A| ≠ 0.

**Method of Finding an Inverse Matrix**

**Some Properties of Inverses**

**a. ****The inverse of the product
of two non-singular matrices is equal to the product of their inverses taken in
reverse order. i.e. If A and B are two non-singular matrices of same order then
(AB) ^{-1} = B^{-1}A^{-1}.**

**b. ****The transpose and inverse
of a non-singular matrix is commutative. i.e. (A ^{-1})^{t} = (A^{t})^{-1}.**

**Worked Out Examples**

*You can comment your questions or problems regarding the inverse
of matrices here.*

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