Q 238 - Intermediate Algebra

Question

In the following exercises, find and then evaluate the indicated minors. 

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

Find the minor $\left(a\right) a_2 \left(b\right) b_2 \left(c\right) c_2$

Step-by-Step Solution

Answer

Part a. The minor of $a_2$ is $2.$

Part b. The minor of $b_2$ is $-4.$

Part c. The minor of $c_2$ is $-2.$

Part a Step 1. Given Information

The given matrix is

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

We have to find the minor of $a_2.$

Part a Step 2. Evaluate the minor of a 2

To evaluate the minor of $a_2$

Let's eliminate the row and column that contains $a_2$

Then write the remaining determinants as $2\times 2$matrix.

Part a Step 3. Solve the matrix

So, the matrix is

$$\begin{gathered} \left|\begin{array}{cc}-3& -4\\ -1& -2\end{array}\right| \\ =(-3)(-2)-(-4)(-1) \\ =6-4 \\ =2 \end{gathered}$$

Thus, the minor of $a_2$ is $2.$

Part b Step 1. Given Information

The given matrix is

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

We have to find the minor of  $b_2.$

Part b Step 2. Evaluate the minor of b 2

To evaluate the minor of $b_2$

Let's eliminate the row and column that contains $b_2$

Then write the remaining determinants as $2\times 2$ matrix.

Part b Step 3. Solve the matrix

So, the matrix is

$$\begin{gathered} \left|\begin{array}{cc}2& -4\\ 0& -2\end{array}\right| \\ =\left(2\right)(-2)-\left(0\right)\left(\_4\right) \\ =-4-0 \\ =-4 \end{gathered}$$

Thus, the minor of $b_2$ is $-4.$

Part c Step 1. Given Information

The given matrix is

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

We have to find the minor of $c_2.$

Part c Step 2. Evaluate the minor of c 2

To evaluate the minor of $c_2$

Let's eliminate the row and column that contains $c_2$

Then write the remaining determinants as $2\times 2$ matrix.

Part c Step 3. Solve the matrix

So, the matrix is

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

Thus, the minor of $c_2$ is $-2.$