# Cofactor matrix and determinant question

• fluidistic
In summary, you need to sum up all the entries of A multipled with their correspondant in the cofactor matrix in order to get the determinant.
fluidistic
Gold Member

## Homework Statement

I have the matrix A as being (-2 3 2; 6 0 3; 4 1 -1) and I'd like to calculate its determinant via calculating its cofactor matrix, even if I know it's much more laborious than just calculating its determinant.
I've calculated its cofactor matrix as being (-3 18 6; 5 -6 14; 9 18 -18). I've checked up the arithmetic twice and even redone all the arithmetic but I always fall over this cofactor matrix so I'm almost sure I didn't make any error.
Now from my notes in order to get the determinant, I must sum up all the entries of A multipled with their correspondant in the cofactor matrix.
That is, the arithmetic should start like : $$(-2)\cdot (-3)+3\cdot 18 + 2\cdot 6$$...
I finally get a result of 216 which is $$3 \times 72$$. While calculating the determinant of A simply, I get 72. I'm sure I'm doing something wrong... and also I don't know if this factor 3 is a pure coincidence.
My question is : is it the right way to calculate the determinant of A, supposing that my cofactor matrix is right?

fluidistic said:
I must sum up all the entries of A multipled with their correspondant in the cofactor matrix.
What exactly do you mean by this?

EDIT: I understand what you did now, what you did was expand along all three rows and then added them up.
You only need to expand along one row. (Doing this will give you 72)

Since you did it for all the rows and then added them up you essentially just added 72+72+72 to give 3x72. That's where the factor of three 3 came in.

fluidistic said:
That is, the arithmetic should start like : $$(-2)\cdot (-3)+3\cdot 18 + 2\cdot 6$$...

That is where the arithmatic should end, as that is 72.

Last edited:
rock.freak667 said:
What exactly do you mean by this?

EDIT: I understand what you did now, what you did was expand along all three rows and then added them up.
You only need to expand along one row. (Doing this will give you 72)

Since you did it for all the rows and then added them up you essentially just added 72+72+72 to give 3x72. That's where the factor of three 3 came in.

That is where the arithmatic should end, as that is 72.
Ah! Thank you so much! Now I understand better my notes, but it wasn't clear at all.

## 1. What is a cofactor matrix?

A cofactor matrix is a square matrix that is formed by finding the cofactors of each element in a given square matrix. The cofactor of an element is the determinant of the minor matrix formed by deleting the row and column containing that element.

## 2. How is a cofactor matrix used to find the determinant of a matrix?

A cofactor matrix is used in the cofactor expansion method to find the determinant of a matrix. The determinant is calculated by multiplying each element in a row or column of the original matrix by its corresponding cofactor in the cofactor matrix, and then adding or subtracting these products according to the pattern of the determinant formula.

## 3. Can a cofactor matrix be used to find the inverse of a matrix?

Yes, a cofactor matrix can be used to find the inverse of a matrix. The inverse of a matrix can be found by dividing the adjugate matrix (transpose of the cofactor matrix) by the determinant of the original matrix.

## 4. What is the relationship between the determinant and the cofactor matrix?

The determinant of a matrix is equal to the sum of the products of each element in a row or column of the matrix with its corresponding cofactor in the cofactor matrix. This relationship is known as the cofactor expansion method for finding determinants.

## 5. How do you calculate the cofactor of an element in a matrix?

The cofactor of an element in a matrix can be calculated by finding the determinant of the minor matrix formed by deleting the row and column containing that element. The sign of the cofactor is determined by the pattern of the determinant formula (alternating between positive and negative).

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