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The matrix is:

$\begin{pmatrix} 3+l & 8 & 3 & 3+l \\ 8 & 9 & 3 & 7 \\ 3 & 3 & 7 & 8 \\ 3+l & 7 & 8 & 13 \end{pmatrix}$

I'm given the above matrix, and I'm asked to figure if it can be the matrix of a linear transformation, for a given $l$.

What is the methodology to find a linear transformation that uses the above as its matrix?

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    I can choose l at will, and chose it to be 0 as i wrote2012-05-16

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As commentators have mentioned, any matrix produces a linear transformation on vectors through multiplication. In detail using $A$ for your matrix, $[w,x,y,z]A=[(8 x+w (3+x)+3 y+(3+x) z, 8 w+9 x+3 y+7 z, 3 w+3 x+7 y+8 z, 7 x+w (3+x)+8 y+13 z)]$

In fact, the same matrix can represent different transformations depending on the basis being used. If you are certain you are supposed to determine if this is a matrix for a given transformation, then you will need to add information about the basis.

In the case Yrogirg is correct about you wanting it to be a nonsingular transformation, then the course of action would be to compute the determinant and see if/when it is zero. You should get $-54\ell^2+501\ell-1167$. By checking the disriminant you can see that it only has two complex roots, so this matrix is always nonsingular (if you are only interested in real matrices.)

I noticed that the matrix is also symmetric, which I thought might come into the picture somehow.

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    Frankly speaking, I still don't understand what Neyo wanted. Neyo, did you solve the problem? If so, post the answer, so people like me could understand the question.2012-05-17