# Easiest way to calculate the determinant of this 4x4 matrix

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!

• use Gauss Elimination Method Commented Jun 17, 2015 at 23:44
• one iteration of Gaussian elimination and then formula for $3 \times 3$ determinant.
Commented Jun 18, 2015 at 0:00

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.

• The last step is certainly wrong. If all the preceding steps are right, then the last entry should be $(61\cdot 43)-(85\cdot32)$, not a fraction. Commented Jun 18, 2015 at 0:17
• You're right, it's such a long time since I did that by hand for the last time... Commented Jun 18, 2015 at 0:19
• Thanks for the answer! Im too tired to fully understand the steps. I'll try tomorrow again :) Commented Jun 18, 2015 at 0:42