Check the determinant of a matrix given a parameter

Question

How do I calculate the determinant of the following matrix?
And for which values of m is the determinant non null?

\begin{bmatrix} 1 & 1 & 1 \\ 2 & m & 3 \\ 4 & m^2 & 9 \end{bmatrix} I have tried the

co-factoring method

and what I got was m(m+1) + 6 I'm trying to figure what would be the right way to do this?

This is when you multiply each element in any row or column by its co factor — Mister Tusk, Nov 11 '18 at 16:29
Oh! Usually, it's called a Laplace expansion along a row or a column. — Bernard, Nov 11 '18 at 16:33
$m=2$ or $n=3$, cf. https://en.wikipedia.org/wiki/Vandermonde_matrix. — Michael Hoppe, Nov 11 '18 at 17:29

F. Knorr · Accepted Answer · 2018-11-11T16:56:08.363

2

In general, you can calculate the determinant of any 3x3 matrix with the method described in the introductory section of the corresponding wikipedia article.

After you have successfully calculated the determinant (-m^2 + 5 m - 6), determine the zeros of this quadratic equation depending on m. Except for these two values of m the determinant will be non-zero.

edited Nov 11 '18 at 16:56

answered Nov 11 '18 at 16:41

F. Knorr

136

score 0 · Answer 2 · answered Nov 11 '18 at 16:23

0

This is a Vandermonde determinant

Hint: subtract the first column from the other two.

answered Nov 11 '18 at 16:23

Bernard

175,478

What do I do after that ? – Mister Tusk Nov 11 '18 at 16:30
Expand the detyerminant along the 1st row. It amounts to calculating a $2\times2$ determinant. – Bernard Nov 11 '18 at 16:31

Check the determinant of a matrix given a parameter

2 Answers2