Question

Describe how to find the minor of 6 in$\left[\begin{array}{ccc}5& 11& 7\\ -1& 3& 8\\ 6& 0& -2\\ & & \end{array}\right]$.

Short answer

The minor of 6 is $\left|\begin{array}{cc}3& 8\\ 0& -2\\ & \end{array}\right|$.

Step-by-step solution

- Minor of an element

The minor of an element is the determinant formed when the row and column containing that element are deleted.

- Find the minor of 6

Apply the definition of the minor of an element. Since 6 is in row 3 column 1 therefore, third row and first column must be deleted and the determinant formed by the other leftover rows and column is the minor of 6.

$\left|\begin{array}{ccc}& & \\ & 3& 8\\ & 0& -2\\ & & \end{array}\right|=\left|\begin{array}{cc}3& 8\\ 0& -2\\ & \end{array}\right|$.

- Calculate the determinant

The determinant of second order matrix is found by calculating the difference of the product of the two diagonals, that is., $\left|\begin{array}{cc}a& b\\ c& d\\ & \end{array}\right|=ad-bc$.

To further simplify multiply the values first and then find the difference.

$\begin{array}{l}\left|\begin{array}{cc}3& 8\\ 0& -2\end{array}\right|=3\times (-2)-8\left(0\right)\\ =-6\end{array}$