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I'm wondering about the approach to another problem.

$B = \left( \begin{array}{ccc} a & 2a & 3a \\ 4b & 5b & 6b \\ 7c & 8c & 9c \end{array} \right) $

and I want to find $|B^4|$ i.e. the determinant of B to the 4th.

I was thinking of getting $B^k = S$\Lambda^k$S^{-1}$ where $S$ is the eigenvector matrix. Is that the only approach, or is there a better way to solve this problem?

$B^T$ provides an interesting matrix that I can manipulate more easily, but I'm not sure that helps me get to what I need.

THanks in advance.

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    The determinant is multiplicative so you only need to compute $\det B$ and this is straightforward using the fact that the determinant is multilinear in the rows. Why is the transpose easier to manipulate?2011-12-13

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There's no need to think about eigenvectors/values for this problem.

Here's a hint. If B' is obtained from $B$ by doing a row operation, then |B'| is a certain multiple of $|B|$. Here are the three types of row operation with their effects on determinants (in each line, B' should be regarded as after the row operation has been performed on $B$).

Row Swap: |B'| = -|B|.

Row Scaling: (multiply a row by a constant $k$). |B'| = k|B|.

Add multiple of one row to a different row: |B'| = |B|.

For your matrix, notice that each row has a particular nice form. Find a series of row operations that relate your matrix to a simpler one. I can see how to obtain a matrix with no unknown constants through row operations. Once you get to that stage, use the fact that $|A^p| = |A|^p$.

Hope this helps!

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    Well there are tons of useful properties, but you'll want to consult a standard text on linear algebra to explore these.2011-12-13