# CBSE Class 12: Determinants - Inverse of a Matrix and its Applications

In this video, we will get to learn about the Inverse of a Matrix and its Applications in solving the simultaneous equation.

- If A is a square matrix then another square matrix denoted by A
^{−1 }is said to be the inverse of A such that A.A^{−1 }= I = A^{−1}.A,^{ }where I^{ }is the identity matrix - A square matrix A is invertible if and only if |A|≠0
- Adjoint of a matrix is defined as the transpose of its cofactor matrix
- To solve a system of equations; the equations can be written in the form of a matrix equation as: AX = B, where:

a) A is a matrix formed by all the coefficients in the system

b) X is a matrix formed by all the variables in the system

c) B is a matrix formed by all the constants in the system

- If |A|≠0, then A
^{−1 }exists and the system of equations is said to be inconsistent having a unique solution - Unique solution of equation AX = B is given by X = A
^{–1}B - If | A| = 0, then we calculate (adj A) B. Here two cases arise:

a) If (adj A) B ≠ O, (where O is a zero matrix), then solution does not exist and the system of equations is called inconsistent

b) If (adj A) B = O, then the system is consistent and have infinitely many solution.