Matrix inverse?

Question

If a 3x3 matrix has zeros on the main diagonal, does it have an inverse?

Answer 1

Your matrix would look like this.

| 0 a b |
| c 0 d |
| e f 0 |

If a matrix is singular (meaning it doesn't have an inverse), then its determinant is zero.

det = 0(0*0 - d*f) - a(c*0 - d*e) + b(c*f - e*0)
= ade + bcf

So it's only singular if ade + bcf = 0. So it just depends.

| 0 1 1 |
| 1 0 1 |
| 1 1 0 | has an inverse. It has a determinant of 2.

| 0 -1 1 |
| 1 0 -1 |
| -1 1 0 | does not. It has a determinant of 0.

Answer 2

A matrix has an inverse iff its determinant is non-zero. Do you know how to calc the det of a 3x3? Use Sarrus' rule.

(zeros on the diagonal don't automatically make the determinant zero) so the answer is 'it can have an inverse'.

It's a good question! Sometimes it helpful to look at the 2x2 case because it's so easy:

0 b
c 0

The determinant is -cb, which is not necessarily zero.

Answer 3

"As a rule of thumb, almost all square matrices are invertible". There are some types of matrices that cannot be inverted, which are called 'singular'.

But that a matrix is singular or not does not depend on it having zeroes in the main diagonal or not.