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The following is a problem from Apostol, Calculus, Volume II (p. 80).

The question requests we compute the determinant of

$$ \begin{pmatrix} a & 1 & 0 & 0 & 0 \\ 4 & a & 2 & 0 & 0 \\ 0 & 3 & a & 3 & 0 \\ 0 & 0 & 2 & a & 4 \\ 0 & 0 & 0 & 1 & a \end{pmatrix}$$

by transforming it to an upper triangular matrix.

Is there an efficient way to transform this to an upper triangular matrix? I can go about row-reducing which should eventually get me there (my attempts at that have lead to extremely long and tedious calculation), but the form of the matrix suggests there might be a faster way? I can't seem to find it. (One can, of course, compute the determinant by just doing the expansion, but I'm more curious if there is a fast way to get to the upper triangular matrix the problem requests.)

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  • $\begingroup$ Maybe an LU-decomposition can help you out? $\endgroup$ – ABC Jun 4 '14 at 21:54
  • $\begingroup$ @ABC Unfortunately, I don't know what that is. This is the first problem set on matrices in the book, so approximately no machinery. It is possible the author just intended an extremely involved algebraic computation. I was hoping otherwise... $\endgroup$ – Bamboo Jun 4 '14 at 21:57
  • $\begingroup$ The main reason why they ask you to transform it into an upper triangular matrix, is that the determinant is just the multiplication of all the elements on the first diagonal. There are some algorithms to transform a matrix, for example the LU-decompostion such that A = LU where L is an lower triangular matrix and U an upper triangular one. For this particular matrix, the determinant should be a(a^2-20a+64) $\endgroup$ – ABC Jun 4 '14 at 22:04
  • $\begingroup$ @ABC Yes, I was just curious if there was a quick path to the transformation to upper triangular that I was missing. I believe the determinant is a(a^4 - 20a^2 + 64), btw. $\endgroup$ – Bamboo Jun 4 '14 at 22:09
  • $\begingroup$ That's right. My bad :-) $\endgroup$ – ABC Jun 4 '14 at 22:10
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This matrix is called a tridiagonal matrix.

In this case, you should be able to turn this matrix into an upper triangular matrix with just four elementary row operations*, all row additions. Row additions don't effect the determinant, so you just have to multiply out the elements on the diagonal of the resulting upper triangular matrix and get the determinant of the result. The standard method of the LU decomposition people are taught does the same thing... there is something called Crout's method for the LU decomposition as well that is easier to calculate out, but it is most likely simpler just to row reduce it.

*it doesn't look like you'll have to do any pivoting at a quick glance. Pivoting just multiplies the determinant by -1.

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  • $\begingroup$ ** unless $a=0$ $\endgroup$ – Algebraic Pavel Jun 4 '14 at 22:58

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