Q 236 - Intermediate Algebra

Question

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

$\left|\begin{array}{ccc}3& -1& 4\\ -1& 0& -2\\ -4& 1& 5\end{array}\right|$

Find the minor $\left(a\right) a_1 \left(b\right) b_2 \left(c\right) c_3$

Step-by-Step Solution

Answer

Part (a) The minor of $a_1$ is $2.$

Part (b) The minor of $b_2$ is $31.$

Part (c) The minor of $c_3$ is $-1.$

Part (a) Step 1. Given Information

The given matrix is

$\left|\begin{array}{ccc}3& -1& 4\\ -1& 0& -2\\ -4& 1& 5\end{array}\right|$

We have to find the minor of $a_1.$

Part (a) Step 2. Evaluate the minor of a 1

To evaluate the minor of $a_1$.

Let's eliminate the row and column that contains $a_1$.

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

Part (a) Step 3. Solve the matrix

So, the matrix is

$$\begin{gathered} \left|\begin{array}{cc}0& -2\\ 1& 5\end{array}\right| \\ =\left(0\right)\left(5\right)-(-2)\left(1\right) \\ =0+2 \\ =2 \end{gathered}$$

Thus, the minor of $a_1$ is $2.$

Part (b) Step 1. Given Information

The given matrix is

$\left|\begin{array}{ccc}3& -1& 4\\ -1& 0& -2\\ -4& 1& 5\end{array}\right|$

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

$$\begin{gathered} \left|\begin{array}{cc}3& 4\\ -4& 5\end{array}\right| \\ =\left(3\right)\left(5\right)-\left(4\right)(-4) \\ =15+16 \\ =31 \end{gathered}$$

Thus, the minor of $b_2$ is $31.$

Part (c) Step 1. Given Information.

The given matrix is

$\left|\begin{array}{ccc}3& -1& 4\\ -1& 0& -2\\ -4& 1& 5\end{array}\right|$

We have to find the minor of $c_3.$

Part (c) Step 2. Evaluate the minor of c 3

To evaluate the minor of $c_3$

Let's eliminate the row and column that contains $c_3$

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

Part (c) Step 3. Solve the matrix

So, the matrix is,

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

Thus, the minor of $c_3$ is $-1.$