Chapter 16.6, Problem 3OE

To determine

To Check: Whether the given matrix has an inverse:

  $\left[\begin{array}{cc}10& 5\\ 9& 4\end{array}\right]$

Answer to Problem 3OE

Yes

Explanation of Solution

Given: $\left[\begin{array}{cc}10& 5\\ 9& 4\end{array}\right]$

Formula Used: $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-cb$

For checking whether a given matrix is invertible it is needed to verify that the determinant of the matrix is non-zero or not.

If the determinant of the given matrix is non-zero then the matrix has an inverse.

If the determinant of the matrix is zero, the matrix does not have an inverse.

Finding the determinant of the given matrix:

  $\left|\begin{array}{cc}10& 5\\ 9& 4\end{array}\right|=(10)(4)-(9)(5)=40-(45)=-5$

The determinant of the given matrix is non-zero so the matrix is invertible.

Yes, the given matrix has an inverse.