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.