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

May 20, 2015 11:25 IST

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–1B
• 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.

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