# Find the determminant?

• Dec 13th 2010, 02:52 PM
Find the determminant?
find the determinant
0 1 -2
-2 2 0
-3 4 1

find the inverse
4 -5
1 -1

find the deterinant
-4 3
5 0
• Dec 13th 2010, 03:39 PM
dwsmith
I expanded across the first row.

$\displaystyle
\begin{vmatrix}
0 & 1 & -2\\
-2 & 2 & 0\\
-3 & 4 & 1
\end{vmatrix}\Rightarrow 0*\begin{vmatrix}
2 & 0 \\
4 & 1
\end{vmatrix}-1*\begin{vmatrix}
-2 & 0 \\
-3 & 1
\end{vmatrix}-2*\begin{vmatrix}
-2 & 2 \\
-3 & 4
\end{vmatrix}=0-1*(-2-0)-2*(-8-(-6))=0+2+4=6$

You should be able to do this one by using 1 and 3 as an outline.

$\displaystyle A^{-1}=\frac{1}{det(A)}*A$

$\displaystyle\begin{vmatrix}
-4 & 3 \\
5 & 0
\end{vmatrix}=-4*0-5*3=0-15=-15$

Definition:
The determinant of an n x n matrix A is defined as

$\displaystyle
det(A) =
\begin{cases}
a_{11}, & \mbox{if } \ n=1 \\
a_{11}A_{11}+a_{12}A_{12}+\dots +a_{1n}A_{1n}, & \mbox{if } \ n>1
\end{cases}
$

where

$\displaystyle A_{1j}=(-1)^{1+j}det(M_{1j}) \ j=1,2,\cdots , n$

are the cofactors associated with the first row expansion.