Solving a 4x4 Matrix Determinant: Factor/Cofactor & Triangular Form

In summary, the conversation revolved around the topic of solving the determinant of a 4x4 matrix. Two methods were discussed: Factor/Cofactor and reduction to triangular form. The individual steps for each method were described, with a focus on the reduction to triangular form being a quicker method. The conversation also touched on the possibility of a matrix having a determinant of 0 and the importance of understanding the process rather than relying solely on a calculator.
  • #1
phantomAI
17
0
I seem to be still having problems with solving the determinant of a 4x4 matrix correctly. There are two methods I can use: Factor/Cofactor and reduction to triangular form.

Ex:
[ 1 2 3 4
-5 2 1 0
6 4 3 2
1 1 1 1]

determinant is 2 based on the calculator

Factor/Cofactor: I did:
1*det[2,1,0; 4,3,2; 1,1,1] - 2*det(-5,1,0;6,3,2; 1,1,1]
+ 3*det[-5,2,0; 6,4,2; 1,1,1] -4*det(-5,2,1; 6,4,3; 1,1,1]

From there I would do the individual det of the 3x3s, but when everything is added up I'm getting a determinant of 0! Did I set things up correctly like on the process of setting up the factor/cofactor method.


With the reduction to triangular form, it looks similar to Guassian elimination where I'm trying to get the matrix reduced to upper triangular form. I know that each row swap I make I need to multiply by (-1). When I'm done I mulitply the terms outside the matrix to the diagonals. However, I'm still a bit confused, so can anyone explain it a bit better than my textbook?

Thanks.
 
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  • #2
From there I would do the individual det of the 3x3s, but when everything is added up I'm getting a determinant of 0!

Which happens to be the correct value... What makes you think a matrix can't have a determinant of 0?
 
  • #3
DOH! I shouldn't always rely on the calculator (or at least me inputting wrong values)


Yeah the Det is 0.

I still have some trouble with the Triangular form method though. This technique is suppose to be quicker than the Factor/Cofactor technique right?
 
  • #4
Yes, a lot quicker. Learn it, it's very useful
 

Related to Solving a 4x4 Matrix Determinant: Factor/Cofactor & Triangular Form

1. What is a matrix determinant?

A matrix determinant is a numerical value that is calculated from the elements of a square matrix. It represents the scaling factor of the matrix and is often used in solving systems of linear equations, finding inverses of matrices, and other mathematical operations.

2. Why is it important to solve a 4x4 matrix determinant?

Solving a 4x4 matrix determinant is important because it allows us to determine whether a system of linear equations has a unique solution, no solution, or infinitely many solutions. It is also an important step in finding the inverse of a 4x4 matrix.

3. What is the factor/cofactor method for solving a 4x4 matrix determinant?

The factor/cofactor method involves breaking down the 4x4 matrix into smaller 3x3 matrices and using specific formulas to find the determinant of each smaller matrix. The determinants are then combined to find the final determinant of the 4x4 matrix.

4. Can the triangular form be used to solve a 4x4 matrix determinant?

Yes, the triangular form can be used to solve a 4x4 matrix determinant. This method involves transforming the 4x4 matrix into an upper or lower triangular matrix and then taking the product of the diagonal elements to find the determinant.

5. Are there any shortcuts or tricks for solving a 4x4 matrix determinant?

There are no shortcuts or tricks for solving a 4x4 matrix determinant. It requires following a specific set of steps and using the appropriate formulas to find the determinant. However, with practice and familiarity, the process can become easier and more efficient.

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