Question

Determine whether each statement is true or false.

18. Some square matrices do not have multiplicative inverses.

Short answer

The given statement is true.

Step-by-step solution

­- Definition of square matrix.

A square matrix is a matrix in which number of rows is equal to number of columns.

­- Definition of inverse of matrix.

A matrixB is said to be an inverse of the matrix Aif$\mathbf{AB}\mathbf{=}\mathbf{BA}\mathbf{=}\mathbf{I}$where I is an identity matrix of the same order as that of matrix Aor B.

­- Description of step.

A square matrixA does not have its inverse if .$\left|A\right|=0$

A square matrixA have its inverse if .$\left|A\right|\ne 0$

Therefore, if the determinant of the matrix is zero then the matrix will not have a multiplicative inverse.

Therefore, some square matrices which have their value of determinant as zero do not have multiplicative inverse.

­- State whether the given statement is true or false.

The given statement is true.