Matrix Proof

Printable View

Nov 1st 2008, 09:10 PM
Hellreaver

Matrix Proof

I need to prove that these determinants are equals.
Note: these are determinants, not matrices, I just don't know how to set them up as determinants using the math code...

$\begin{bmatrix}a1+b1t&a2+b2t&a3+b3t\\a1t+b1&a2t+b2 &a3t+b3\\c1&c2&c3 \end{bmatrix}$ = (1- $t^2$ ) $\begin{bmatrix}a1&a2&a3\\b1&b2&b3\\c1&c2&c3 \end{bmatrix}$

This tells me that I need to somehow get (1- $t^2$ ) multiplied into the determinant. I know that to get that out, one would have to multiply a row by (1- $t^2$ ), or possibly (1-t) on two different rows... Am I thinking it the right direction?
Nov 1st 2008, 10:11 PM
mr fantastic

Quote:

Originally Posted by Hellreaver

I need to prove that these determinants are equals.
Note: these are determinants, not matrices, I just don't know how to set them up as determinants using the math code...

$\begin{bmatrix}a1+b1t&a2+b2t&a3+b3t\\a1t+b1&a2t+b2 &a3t+b3\\c1&c2&c3 \end{bmatrix}$ = (1- $t^2$ ) $\begin{bmatrix}a1&a2&a3\\b1&b2&b3\\c1&c2&c3 \end{bmatrix}$

This tells me that I need to somehow get (1- $t^2$ ) multiplied into the determinant. I know that to get that out, one would have to multiply a row by (1- $t^2$ ), or possibly (1-t) on two different rows... Am I thinking it the right direction?

Consider the matrix $A = \begin{bmatrix}a1&a2&a3\\b1&b2&b3\\c1&c2&c3 \end{bmatrix}$ .

Apply the row operation R1 --> R1 + t R2 on matrix A to get matrix B: det(B) = det(A).

Apply the row operation (1 - t^2) R2 + R1 --> R2 on matrix B to get matrix C: det(C) = (1 - t^2) det(B) ......
Nov 1st 2008, 10:18 PM
Hellreaver

Quote:

Originally Posted by mr fantastic

Consider the matrix $A = \begin{bmatrix}a1&a2&a3\\b1&b2&b3\\c1&c2&c3 \end{bmatrix}$ .

Apply the row operation R1 --> R1 + t R2 on matrix A to get matrix B: det(B) = det(A).

Apply the row operation (1 - t^2) R2 + R1 --> R2 on matrix B to get matrix C: det(C) = (1 - t^2) det(B) ......

Ok, that's tricky. Thank you so much!