Chapter 2, Problem 1SE

In Exercises 1–8, evaluate the determinant of the given matrix by (a) cofactor expansion and (b) using elementary row operations to intro-duce zeros into the matrix.

$\left[\begin{array}{cc}-4& 2\\ 3& 3\end{array}\right]$

To determine

The determinant of the matrix $A=\left[\begin{array}{cc}-4& 2\\ 3& 3\end{array}\right]$, using cofactor expansion.

Answer to Problem 1SE

The determinant of the matrix $A=\left[\begin{array}{cc}-4& 2\\ 3& 3\end{array}\right]$ is $\underset{\_}{-18}$.

Explanation of Solution

Definition used:

Determinant:

“The determinant of a matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is given by the formula $ad-bc$ and the matrix A is said to be invertible if, $ad-bc\ne 0$.”

Calculation:

The given matrix is $A=\left[\begin{array}{cc}-4& 2\\ 3& 3\end{array}\right]$.

Use cofactor expansion and obtain the value of the determinant as shown below.

$\begin{array}{c}\det \left(A\right)=\left(-4\right)\cdot 3-2\cdot 3\\ =-12-6\\ =-18\end{array}$

Thus, the determinant of the matrix $A=\left[\begin{array}{cc}-4& 2\\ 3& 3\end{array}\right]$ is $\underset{\_}{-18}$.

To determine

The determinant of the matrix $A=\left[\begin{array}{cc}-4& 2\\ 3& 3\end{array}\right]$, using elementary row operations.

Answer to Problem 1SE

The determinant of the matrix $A=\left[\begin{array}{cc}-4& 2\\ 3& 3\end{array}\right]$ is $\underset{\_}{-18}$.

Explanation of Solution

The given matrix is $A=\left[\begin{array}{cc}-4& 2\\ 3& 3\end{array}\right]$.

Obtain the value of the determinant as shown below.

$\begin{array}{c}\left|A\right|=\left|\begin{array}{cc}-4& 2\\ 3& 3\end{array}\right|\left(R_2\to \frac{R_2}{3}\right)\\ ~3\cdot \left|\begin{array}{cc}-4& 2\\ 1& 1\end{array}\right|\left(R_1\to R_1+4R_2\right)\\ ~3\cdot \left|\begin{array}{cc}0& 6\\ 1& 1\end{array}\right|\left(R_2\to R_2-\frac{1}{6}{R}_1\right)\\ ~3\cdot \left|\begin{array}{cc}0& 6\\ 1& 0\end{array}\right|\end{array}$

On further simplifications,

$\begin{array}{c}\left|A\right|=3\cdot \left(-6\right)\\ =-18\end{array}$

Thus, the determinant of the matrix $A=\left[\begin{array}{cc}-4& 2\\ 3& 3\end{array}\right]$ is $\underset{\_}{-18}$.