Q18. - Algebra 2

Question

Determine whether each statement is true or false.

18. Some square matrices do not have multiplicative inverses.

Step-by-Step Solution

Answer

The given statement is true.

Step 1 ­- Definition of square matrix.

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

Step 2 ­- 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.

Step 3 ­- 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.

Step 4 ­- State whether the given statement is true or false.

The given statement is true.