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

Short 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$-2.$

Step-by-step solution

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 a2

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 ofrole="math" localid="1644566412877" $b_2.$

Part b Step 2. Evaluate the minor of b2 

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 c2 

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$-2.$