Matrix Determinants: Find x for Invertibility

[SherlockOhms]

Homework Statement

For which values of x is the matrix (see attachment) invertible?

Homework Equations

Row ops. Cofactors etc..

The Attempt at a Solution

Well, a matrix is only invertible when it's determinant is non zero. I've begun doing some row ops and have just hit a little snag. If you look at the attachment you'll see that I can facto (1 - x) out from the minor matrix. I remember hearing in a lecture that you have to factor out (1 - x) from both the top and bottom row of the matrix (i.e. you'll have (1 -x)^2 factored out instead of just (1 - x). Could somebody explain why you don't just factor out (1 - x) one like you would with factoring a scalar out of a matrix as normal?x

 

[SherlockOhms]

 

[SteamKing]

In the reduced 2x2 matrix, the factor (1-x) is common to all of the members of the matrix. You can only factor it out once. Whatever you heard about factoring the matrix was incorrect.

See this article: http://en.wikipedia.org/wiki/Matrix_(mathematics)

specific topic: scalar multiplication ofa matrix

 

[SherlockOhms]

Thanks for clearing that up.

 

[verty]

Even with a scalar, wikipedia confirms that to scale a row by m scales the determinant by m, which is clear if you think of the formula for the determinant.

 

[SherlockOhms]

Yeah. That's actually what got me thinking about it in the first place.