# Cofactor Expansion: Find Determinant of 4x3 Matrix

• fk378
In summary, the conversation was about finding the determinant of a 3x3 matrix and the confusion over getting different answers when expanding by different rows. The correct answer is -1, which can be obtained by expanding by the first or third column. The confusion was due to not multiplying one of the terms in the cofactor expansion.
fk378

## Homework Statement

Find the determinant of
4...3...-5
5...2...-3
0..-1...2

## The Attempt at a Solution

I've tried to get the answer using each of the 3 rows and each time I get a different answer. For the first row I get 14, the second row I get -31, and the third row I get -1. However, I thought that the determinant would be the same value for whichever row (or column) you choose to expand.

fk378 said:

## Homework Statement

Find the determinant of
4...3...-5
5...2...-3
0..-1...2

## The Attempt at a Solution

I've tried to get the answer using each of the 3 rows and each time I get a different answer. For the first row I get 14, the second row I get -31, and the third row I get -1. However, I thought that the determinant would be the same value for whichever row (or column) you choose to expand.
Yes, it certainly should be!

Expanding by the first column (since it has only 2 non-zero entries):
$$\left|\begin{array}{ccc}4 & 3 & -5 \\ 6 & 2 & -3 \\ 0 & -1 & 2 \end{array}\right|= 4\left|\begin{array}{cc}2 & -3 \\ -1 & 2\end{array}\left|- 5\right|\begin{array}{cc} 3 & -5 \\ -1 & 2\end{array}\right|$$
$$4(4- 3)- 5(6- 5)= 4(1)- 5(1)= -1$$
The determinant is -1. I notice that is what you got using the third row which also has only two non-zero entries. Perhaps it is that third entry that is confusing you.

But if you try to work out the determinant using the other rows that do not contain the 0 entry, it does not come out to -1! Why?

Ah, I just realized I wasn't multiplying one of the terms with the cofactor expansion! I feel silly...thanks for your help though!

## 1. What is a cofactor expansion?

A cofactor expansion is a method used to find the determinant of a square matrix. It involves breaking down the matrix into smaller submatrices and using algebraic operations to calculate the determinant.

## 2. Why is the cofactor expansion method used?

The cofactor expansion method is used because it is a systematic and efficient way to find the determinant of a matrix. It is also useful for solving systems of linear equations and finding the inverse of a matrix.

## 3. How do you perform a cofactor expansion on a 4x3 matrix?

To perform a cofactor expansion on a 4x3 matrix, you first choose a row or column to expand along. Then, for each element in that row or column, you find the cofactor (the determinant of the submatrix formed by removing the row and column that the element is in). Finally, you multiply each cofactor by its corresponding element and add them together to get the determinant of the original matrix.

## 4. Can a cofactor expansion be used on any size matrix?

Yes, a cofactor expansion can be used on any square matrix. However, as the matrix gets larger, the calculation becomes more complex and time-consuming.

## 5. Are there any shortcuts or tricks for performing a cofactor expansion?

Yes, there are a few shortcuts and tricks that can make a cofactor expansion easier. For example, you can use the Laplace expansion theorem to simplify the calculation by choosing a row or column with many zeros. You can also use the properties of determinants, such as the fact that multiplying a row or column by a constant also multiplies the determinant by that constant.

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