Chapter 3.6, Problem 45E

To determine

To find: the value of determinant for the given matrix; and state whether the inverse of this matrix exists.

Answer to Problem 45E

Determinant is $-1$ .

Yes the inverse for this matrix exist.

Explanation of Solution

Given information:

  $\left[\begin{array}{cc}\begin{array}{l}1\\ 2\end{array}& \begin{array}{l}3\\ 5\end{array}\end{array}\right]$

Calculation:

The value of given determinant can be evaluated as follows:

  $\begin{array}{c}\left|\begin{array}{cc}\begin{array}{l}1\\ 2\end{array}& \begin{array}{l}3\\ \text{5}\end{array}\end{array}\right|=\left[1\left(5\right)\right]-\left[2\left(3\right)\right]\\ =5-6\\ =-1\end{array}$

Thus, the determinant is $-1$ .

Yes the inverse for given matrix exists.

For $A=\left[\begin{array}{cc}\begin{array}{l}a\\ c\end{array}& \begin{array}{l}b\\ d\end{array}\end{array}\right]$ , the inverse formula is $A^{-1}=\frac{1}{\det }\left[\begin{array}{cc}\begin{array}{l}d\\ -c\end{array}& \begin{array}{l}-b\\ a\end{array}\end{array}\right]$ . Put the values in the above formula and evaluate the inverse for the given matrix as follows:

  $\begin{array}{c}{A}^{-1}=\frac{1}{-1}\left[\begin{array}{cc}\begin{array}{l}5\\ -2\end{array}& \begin{array}{l}-3\\ 1\end{array}\end{array}\right]\\ =\left[\begin{array}{cc}\begin{array}{l}-5\\ 2\end{array}& \begin{array}{l}3\\ -1\end{array}\end{array}\right]\end{array}$

Thus, the inverse for given matrix is $\left[\begin{array}{cc}\begin{array}{l}-5\\ 2\end{array}& \begin{array}{l}3\\ -1\end{array}\end{array}\right]$ .