Q 277 - Intermediate Algebra

Question

Explain what is meant by the minor of an entry in
a square matrix.

Step-by-Step Solution

Answer

The minor of an entry in a $3\times 3$ determinant is the $2\times 2$ determinant found by eliminating the row and column in the $3\times 3$ determinant contains the entry.

Step 1. To find

To explain the minor of an entry in a square matrix.

Step 2. Minor of an entry in a square matrix

The minor of an entry in a $3\times 3$ determinant is the  $2\times 2$ determinant found by eliminating the row and column in the $3\times 3$ determinant containing the entry.

Step 3. Minor of an entry a 1

Let us consider the $3\times 3$ square matrix $\left[\begin{array}{ccc}3& 2& 3\\ 6& 1& 5\\ 2& 1& 2\end{array}\right]$

To evaluate the minor of an entry $a_1$, eliminate the first row and first column.

Minor of $a_1=\left|\begin{array}{cc}1& 5\\ 1& 2\end{array}\right|$

                   $=2-5$

                   $=-3$

The minor of $a_1$ is $-3$

Step 2. Minor of an entry b 2

To evaluate the minor of an entry $b_2$,

eliminate the second row and second column.

Minor of $b_2=\left|\begin{array}{cc}3& 3\\ 2& 2\end{array}\right|$

                   $=3\left(2\right)-3\left(2\right)$

                   $=6-6$

                   $=0$

The minor of $b_2$ is $0$

Step 5. Minor of an entry c 3

To evaluate the minor of an entry $c_3$, eliminate the third row and third column.

$c_3=\left|\begin{array}{cc}3& 2\\ 6& 1\end{array}\right|$

    $=3\left(1\right)-2\left(6\right)$

    $=3-12$

    $=-9$

The minor of $c_3$ is $-9$