Chapter 8.3, Problem 2E

To determine

To fill the blank space: A matrix that has an inverse is invertible or .......... A matrix that does not have an inverse is -------------.

Answer to Problem 2E

A matrix that has an inverse is invertible or …nonsingular………….. A matrix that does not have an inverse is ----singular---------.

Explanation of Solution

Given information:

Given invertble and non invertible matrices

.

A matrix is said to be invertible, if its determinant is not equal to zero. As we know that the matrix with determinant not equal to zero is called nonsingular matrix.

Therefore, the invertible matrices also called as non singular matrix.

Example : The identity matrix is

  $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ invertible as its determinant is $1\ne 0$ ,

Thus matrix is invertible and hence identity matrix is non-singular matrix

Similar way a matrix is said to be non-invertible, if its determinant is equal to zero. As we know that the matrix with determinant equal to zero is called singular matrix.

Therefore, the non-invertible matrices also called as singular matrix.

Example : The zero matrix is

  $O=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$ non-invertible as its determinant is $0$ ,

Thus matrix is non- invertible and hence zero matrix is singular matrix