Thread: Matrix doing a cofactor expansion

Evaluate the determinant of the following matrix A =
$\displaystyle
\left|\begin{array}{ccccc}2 & 1 & -3 & -1 & 2 \\ 0 & -1 & 0 & 3 & 1 \\ 0 & 2 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 2 \\ 0 & 3 & 0 & 3 & 2 \end{array}\right|
$

by doing a co-factor expansion.

Please expalin how to do it, thank you!
this is the exercise on my task

If you have:

$\displaystyle \begin{array}{lcr}
\mbox a & b & c \\
\mbox d & e & f \\
\mbox g & h & i\end{array}
$

The cofactor for $\displaystyle a$ is the determinant of:

$\displaystyle \begin{array}{lcr}
\mbox e & f \\
\mbox h & i \end{array}
$

The cofactor for $\displaystyle b$ is negative(-) of the determinant of:

$\displaystyle \begin{array}{lcr}
\mbox d & f \\
\mbox g & i \end{array}
$

The cofactor for $\displaystyle c$ is the determinant of:

$\displaystyle \begin{array}{lcr}
\mbox d & e \\
\mbox g & h \end{array}
$

Why are you being asked to do a "co-factor expansion" if you don't know what that is?

I would recommend starting your co-factor expansion using the first column. That way you just have 2 times $\displaystyle \left|\begin{array}{cccc}-1 & 0 & 3 & 1 \\ 2 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 3 & 0 & 3 & 2\end{array}\right|$ and expand that determinant on the second column. Do you see why?

Originally Posted by HallsofIvy

Why are you being asked to do a "co-factor expansion" if you don't know what that is?

I would recommend starting your co-factor expansion using the first column. That way you just have 2 times $\displaystyle \left|\begin{array}{cccc}-1 & 0 & 3 & 1 \\ 2 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 3 & 0 & 3 & 2\end{array}\right|$ and expand that determinant on the second column. Do you see why?

I am sorry, I am not good on maths , can you explain more please !!!
I don't understand what is "just have 2 times" mean?
Is it mean 2 times every number 1 & 0 & 3 & 1 \\ 2 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 3 & 0 & 3 & 2 to evaluate matrix A?