TITLE: Easiest way to calculate the determinant of this 4x4 matrix
QUESTION [2 upvotes]: I have this 4x4 matrix:
$$A=
        \begin{pmatrix}
        2 & 3 & 1 & 0 \\
        4 & -2 & 0 & -3\\
        8 & -1 & 2 & 1\\
        1 & 0 & 3 & 4\\
        \end{pmatrix}
$$
I read that it's easy to calculate it by converting the matrix to upper diagonal. I tried that using line operations but I couldn't make it upper diagonal. Is this the best/easiest method? If so can anyone help me with the process?
Then I have to calculate the eigenvalues and eigenvectors. Any suggestion on how to find them? Do I have to calculate the det(A -λI) to find the characteristic equation? Is there an easy way to find it for such a matrix like A?
Any help will be much appreciated.
Thanks in advance!

REPLY [3 votes]: As suggested in the comments, Gauss elimination is usually the way to go, and the fastest in this case, too:
$$\det A=
        \det\begin{pmatrix}
        2 & 3 & 1 & 0 \\
        4 & -2 & 0 & -3\\
        8 & -1 & 2 & 1\\
        1 & 0 & 3 & 4\\
        \end{pmatrix}
=
        \det\begin{pmatrix}
        0 & 3 & -5 & -8 \\
        0 & -2 & -12 & -19\\
        0 & -1 & -22 & -31\\
        1 & 0 & 3 & 4\\
        \end{pmatrix}
=
        (-1)^{4+1}\cdot 1\cdot\det\begin{pmatrix}
        3 & -5 & -8 \\
        -2 & -12 & -19\\
        -1 & -22 & -31\\
        \end{pmatrix}
=
        -\det\begin{pmatrix}
        0 & -71 & -101 \\
        0 & 32 & 43\\
        -1 & -22 & -31\\
        \end{pmatrix}
=
        -1\cdot(-1)^{3+1}\cdot(-1)\cdot\det\begin{pmatrix}
        -71 & -101 \\
        32 & 43\\
        \end{pmatrix}
= (-71)\cdot 43-(-101)\cdot 32=179
$$
(Wolfram Alpha-verified result; I never could remember the 3x3-formula, so I don't use it)
If you absolutely want an upper diagonal matrix, you can do this, but it's only a restriction of the normal algorithm:
$$\det A=
        \det\begin{pmatrix}
        2 & 3 & 1 & 0 \\
        4 & -2 & 0 & -3\\
        8 & -1 & 2 & 1\\
        1 & 0 & 3 & 4\\
        \end{pmatrix}
=
        \det\begin{pmatrix}
        2 & 3 & 1 & 0 \\
        0 & -8 & -2 & -3\\
        0 & -13 & -2 & 1\\
        0 & -\frac12 & \frac52 & 4\\
        \end{pmatrix}
=
        \det\begin{pmatrix}
        2 & 3 & 1 & 0 \\
        0 & -8 & -2 & -3\\
        0 & 0 & ? & ?\\
        0 & 0 & ? & ?\\
        \end{pmatrix}
=
        \det\begin{pmatrix}
        2 & 3 & 1 & 0 \\
        0 & -8 & -2 & -3\\
        0 & 0 & ? & ?\\
        0 & 0 & 0 & ?\\
        \end{pmatrix}
$$
(I'm too lazy to calculate the $?$ now, just continue with the Gaussian Elimination. The determinant will then be the product of the entries on the diagonal.)
For the eigenvalues: yes, you have to calculate the characteristic polynomial.
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