Question f8dbd

Mar 13, 2017

Minor: $| \left(5 , 4\right) , \left(4 , - 1\right) | \textcolor{w h i t e}{\text{XXX}}$Co-factor: $- | \left(5 , 4\right) , \left(4 , - 1\right) |$

Explanation:

The minor of a determinant is created by deleting all elements in the same row or column as the selected element:
|(5,4,color(red)(cancel(color(black)2))), (color(red)(cancel(color(black)(1))),color(red)(cancel(color(black)(7))),color(red)(cancel(3))), (4,-1,color(red)(cancel(color(black)(1))))| =|(5,4),(4,-1)|#

The co-factor of a determinant is created by multiplying the minor by ${\left(- 1\right)}^{r + c}$ where $r$ is the row number of the selected element and $c$ is the column number of the selected element.
In this case $\textcolor{red}{3}$ is in row $2$ column $3$.
(The top left is always row $1$, column $1$)
So for this example
$\textcolor{w h i t e}{\text{XXX}} {\left(- 1\right)}^{r + c} = {\left(- 1\right)}^{2 + 3} = {\left(- 1\right)}^{5} = - 1$
and the co-factor is
$\textcolor{w h i t e}{\text{XXX}} - 1 \cdot | \left(5 , 4\right) , \left(4 , - 1\right) |$