Determinant

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October 13th 2012, 11:43 AM
Tala

Determinant

How do I find the determinant for this matrix:

first row: (x+y 3 2 1)
second row: ( 0 1 0 0)
third row: ( 0 2 x+y 3)
fourth row: ( 0 3 0 1)

(x,y) ∈ R^{2

Any ideas ?}
October 13th 2012, 12:48 PM
Soroban

Re: Determinant

Hello, Tala!

Are you waiting for Divine Intervention?
You haven't been taught to evaluate determinants in general?

Quote:

$\text{Evluate: }\:\begin{vmatrix}\:x+y & 3 & 2 & 1\: \\ 0 & 1 & 0 & 0\: \\ 0 & 2 & x+y & 3\: \\ 0 & 3 & 0 & 1\: \end{vmatrix}$

Reading down the first column and using cofactors:

$D \;=\;(x+y)\begin{vmatrix}1&0&0\\2&x\!+\!y&3\\3&0&1 \end{vmatrix} \,-\, 0\begin{vmatrix}3&2&1\\2&x\!+\!y&3\\3&0&1 \end{vmatrix} \,+\, 0\begin{vmatrix}3&2&1\\1&0&0\\3&0&1\end{vmatrix} \,-\, 0\begin{vmatrix}3&2&1 \\ 1&0&0 \\ 2&x\!+\!y&3 \end{vmatrix}$

$D \;=\;(x+y)\big[1(x+y-0) - 0 + 0\big] \:-\: 0 \:+\: 0 \:-\: 0$

$D \;=\;(x+y)(x+y)$

$D \;=\;(x+y)^2$
October 13th 2012, 01:42 PM
Tala

Re: Determinant

No I actually haven't been taught to evaluate determinants in general only by Gauss-Jordan elimination. I do know how to find the determinant and I got the same result as you. I'm just wondering if there's anything smart I can do with the function ?
October 13th 2012, 02:12 PM
HallsofIvy

Re: Determinant

Okay, by Gauss-Jordan elimination.

Since the first column is $\begin{bmatrix}x+y \\ 0 \\ 0 \\ 0\end{bmatrix}$ , we don't have to do anything to the first column. To get "0"s below the diagonal in the second column, subtract twice the second row from the third row and subtract three times the second row from the fourth row:
$\begin{bmatrix}x+ y & 3 & 2 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & x+ y & 3\\ 0 & 0 & 0 & 1\end{bmatrix}$

Since that is now a triangular matrix, we can practically "read" the determinant of the diagonal. That was easy, wasn't it?
October 13th 2012, 02:25 PM
Tala

Re: Determinant

So the function (x+y)² is the determinant ?