Inverse of a Matrix

If A and B are non-singular square matrices of the same order such that AB = BA = I (where I is the identity matrix of the same order as A and B), then A and B are called inverses of each other.

We write A⁻¹ = B and B⁻¹ = A.

i.e. AA⁻¹ = A⁻¹A = I.

• If |A| ≠ 0, then A⁻¹ exists.
• If the inverse of a square matrix exists, then it is unique. A matrix can not have more than one distinct inverse.

By Adjoint Method: \[A^{-1}=\frac{\mathrm{adj}A}{|A|}\]

By Using Algebraic Equation: A matrix A and an algebraic equation in matrix A is in the form of A² + bA + C = O.

\[A^{-1}=\frac{1}{C}\left[-aA-bI\right]\]