Find the determminant?

• Dec 13th 2010, 01:52 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, 02:39 PM
dwsmith
I expanded across the first row.

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

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

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

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

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

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

where

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

are the cofactors associated with the first row expansion.