Chapter 4.1, Problem 61E

To determine

determinant for $\left[\begin{array}{cc}-15& 5\\ -9& 3\end{array}\right]$ .and tell whether an inverse exists for the matrix.

Answer to Problem 61E

  $a=\pm 1,a=\pm \left(\sqrt{2}\right)i$

Explanation of Solution

Given information: Find the determinant for $\left[\begin{array}{cc}-15& 5\\ -9& 3\end{array}\right]$ . Tell whether an inverse exists for the matrix

Calculation:

Factorize the equation

  $\left(a^2-1\right)\left(a^2+2\right)=0$

Factorizing further shows that of the 4 roots 2 are real and 2 are complex

  $\left(a-1\right)\left(a+1\right)\left(a^2+2\right)=0$

Real roots

  $\begin{array}{l}a=\pm 1\\ a^2=-2\end{array}$

Complex roots

  $a=\pm \left(\sqrt{2}\right)i$

The graph is a parabola intersecting the x-axis at 1 and (- 1) and having a minimum value.