# Determinant of a 4x4 Matrix Stuck

1. Jun 19, 2012

### erodger

1. The problem statement, all variables and given/known data
Hello, I am stuck on this particular question for my homework.

It is a 4x4 Matrix that consists of

a b b b

b a b b

b b a b

b b b b

3. The attempt at a solution

My approach has been to factor out the a to give the first row of 1 b b b and then use that to make the first column consist of

1
0
0
0

and so i can then do the cofactor expansion and reduce it to a 3x3 matrix. but after that step, it gets extremely tedious and i believe that the approach must be wrong.

Can anyone point out a simpler approach to this question or was I on the right track and just have to endure the tedious algebra?

2. Jun 19, 2012

### pwsnafu

How did you do that? If you factor out a, you get $1 \quad \frac{b}{a} \quad \frac{b}{a} \quad \frac{b}{a}$

Hint: What do you know about block matrices? Specifically, if each block is 2x2 and you have 2x2 of them?

3. Jun 19, 2012

### ehild

Why not subtracting the fourth row from all others?

ehild

4. Jun 19, 2012

### erodger

yeah that fourth row subtraction may have been the best... oh well i solved it by converting that into 4 3x3 matrices and then solving all of those 3x3 matrices.

thanks for the replies though.

5. Jun 19, 2012

### ehild

Subtracting the fourth raw from all other rows, you get the determinant

(a-b) 0 0 0
0 (a-b) 0 0
0 0 (a-b) 0
b b b b

Expand with respect to the fourth column.

ehild

6. Jun 19, 2012

### HallsofIvy

Staff Emeritus
In fact, because that is a "lower triangular" matrix, its determinant is just the product of the numbers on the main diagonal.

In any case, your suggestion that he subtract the last row from each of the other rows was excellent.

7. Jun 19, 2012

### ehild

I knew it but I could not tell the complete solution.

ehild

8. Jun 19, 2012

### HallsofIvy

Staff Emeritus
Well, he still has to do the multiplication!:tongue: