Q. 388 - Intermediate Algebra

Question

In the following exercise, find and then evaluate the indicated minors

$\left|\begin{array}{rrr}-1& -3& 2\\ 4& -2& -1\\ -2& 0& -3\end{array}\right|$

Find the minor

a. $a_1$ b. $b_1$ c. $c_2$

Step-by-Step Solution

Answer

Part (a). The minor of $a_1$ is $\left|\begin{array}{cc}-2& -1\\ 0& -3\end{array}\right|$

$\left|a_1\right|=6$

Part (b). The minor of $b_1$ is $\left|\begin{array}{cc}4& -1\\ -2& -3\end{array}\right|$

$\left|b_1\right|=-14$

Part (c). The minor of $c_2$ is $\left|\begin{array}{cc}-1& -3\\ -2& 0\end{array}\right|$

$\left|c_2\right|=-6$

Step 1. Finding and evaluating the minors a 1 .

To find minors we have to remove the rows and columns containing that element.

here, $a_1=-1$.

Thus the minor of $a_1$ is:

$\left[\begin{array}{cc}-2& -1\\ 0& -3\end{array}\right]$

$$\begin{gathered} \left|a_1\right|=\left|\begin{array}{cc}-2& -1\\ 0& 3\end{array}\right| \\ \left|a_1\right|=(-2\times -3)-(-1\times 0) \\ \left|a_1\right|=6 \end{gathered}$$

Step 2. Finding and evaluating the minors b 1

To find minors we have to remove the rows and columns containing that element.

here, $b_1=-3$

thus, the minors of $b_1$:

$\left|\begin{array}{cc}4& -1\\ -2& -3\end{array}\right|$

$$\begin{gathered} \left|b_1\right|=4\times (-3)-(-2)\times (-1) \\ \left|b_1\right|=-12-2 \\ \left|b_1\right|=-14 \end{gathered}$$

Step 3. Finding and evaluating the minors c 2

To find minors we have to remove the rows and columns containing that element.

here, $c_2=-1$

The minor of $c_2$ is:

$\left|\begin{array}{cc}-1& -3\\ -2& 0\end{array}\right|$

$$\begin{gathered} \left|c_2\right|=-1\times 0-(-3\times -2) \\ \left|c_2\right|=-6 \end{gathered}$$