Chapter 7, Problem 141CR

To determine

The conditions for the existence of inverse of a given matrix.

Answer to Problem 141CR

The inverse of a matrix exists if-

1. The matrix is a square matrix
2. The matrix is nonsingular.

Explanation of Solution

A matrix is invertible if and only if it is a nonsingular square matrix. Consider a matrix A of order $m\times n$ and then its inverse matrix Bwill be of order $n\times m$ . Then AB and BA will have different dimensions and so cannot be equal to each other which is a contradiction since if $B=A^{-1}$ then $AB=BA=I$ must be true. Hence, A cannot have different number of rows and columns, viz. A must be a square matrix.

Now, the inverse of matrix can also be found using the formula-

  $A^{-1}=\frac{adj(A)}{\left|A\right|}$ .

If A is a singular matrix then $\left|A\right|=0$ and from the above relation, $A^{-1}$ becomes undefined.

Hence, for an inverse to exist the matrix should also be nonsingular, i.e.- $\left|A\right|\ne 0$ .

Thus, a matrix is invertible if and only if it is a nonsingular square matrix.