# Determinant of a 4x4 Matrix

## Homework Statement

The matrix is in the following form

2 4 -2 -2
1 3 1 2
1 3 1 3
-1 2 1 2

## The Attempt at a Solution

I subtracted equation 2 from equation 3 and came up with the following matrix

2 4 -2 -1
1 3 1 2
0 0 0 1
1 -2 1 2

It seems to make sense to pivot around the 1 in the third equation. I tried it and got 5 which I know is not that answer using an online Matrix calculator and my professor.

Dick
Homework Helper
Nothing wrong with what you've done so far. How did you get the rest of the way to '5'?

I tried to apply the 3x3 matrix times the value of 1. I use the diagonal method. I am not sure what it is actually called.

I have some confusion between column and row evaluations. I am not sure which one to use (row seems to make sense since there are several 0's).

Use row operations to bring the matrix to diagonal form. The determinant is the sum of the diagonals. Also, remember what the operations do to the determinant. Multiplying any row by a constant multiplies the determinant by a constant. Exchanging rows makes the determinant negative. Adding rows does nothing.

Dick
Homework Helper
I tried to apply the 3x3 matrix times the value of 1. I use the diagonal method. I am not sure what it is actually called.

It looks like you are trying to do "expansion by minors" http://mathworld.wolfram.com/DeterminantExpansionbyMinors.html and working along the third row is fine. That would actually give you (-1) (there's a +/- sign for every matrix position) times a 3x3 matrix. But the determinant of the 3x3 matrix isn't 5 either.

Using row exchanges and factoring I came up with the following matrix.

1 2 -1 -1
1 3 1 2
0 0 0 1
-1 2 1 24

The example you gave me showed doing the pivot on a 3x3. Where do I pivot on the 4x4?

I was thinking that we do the 3x3 matrix evaluation 4 times but where do I begin.

Never mind. I factored it out completed and with subtraction of equations I came up with a line of all 0's so there is no determinant. I entered it wrong above. That cant be right. What I have left is this.

1 0 -1 -3
0 1 0 0
0 0 0 1
0 0 0 0

Dick
Homework Helper
Using row exchanges and factoring I came up with the following matrix.

1 2 -1 -1
1 3 1 2
0 0 0 1
-1 2 1 24

The example you gave me showed doing the pivot on a 3x3. Where do I pivot on the 4x4?

I was thinking that we do the 3x3 matrix evaluation 4 times but where do I begin.

Yes, you do the 3x3 four times, except you don't have to evaluate the 3x3 four times, because if you are running along the third row three of the 3x3 matrices will have a zero factor in front. If you reduced to matrix to get a full row of zeros, that's certainly wrong. The determinant of the matrix isn't zero.

Do I base the 3x3 matrices based on using the top of each column for the multiplier?

Dick
Homework Helper
Do I base the 3x3 matrices based on using the top of each column for the multiplier?

The multipliers are the elements of the row you are using to expand.

I tried that on the row with mostly 0's to simplify the problem but my answer came up wrong. Could you post step by step how to do it. You don't need to do the actual calculation, just the steps. I want to understand the concept, not necessarily the answer. Perhaps with the original matrix.

Last edited:
Dick
Homework Helper
I tried that on the row with mostly 0's to simplify the problem but my answer came up wrong. Could you post step by step how to do it. You dont need to do the actual calculation, just the steps. I want to understand the concept, not necessarily the answer.

You need to show your wrong calculation in detail. We can't figure out what are doing wrong until you show us what it is.

You need to show your wrong calculation in detail. We can't figure out what are doing wrong until you show us what it is.

I figured it out. I went ahead and reread the material for a third and worked out exactly what they had for an example and I saw exactly were I was going wrong. I think I was trying to combine row interactions with factoring of columns. I am reading now how I can do it with any row or any column. Thank you for the help.