# Determinants from any row or column

• Fan de Douze
In summary, the conversation discusses the rule for calculating a determinant by using cofactors and entries of a row or column of a matrix. The speaker mentions a negative factor that appears in their calculations and realizes that it is due to not considering the (-1)^(i+j) factor in the cofactor. The conversation concludes with the understanding that this factor is necessary for correctly calculating the determinant.
Fan de Douze
I'm having a problem with this rule in general. Apparently one can calculate the determinant by multiplying the cofactors and entries of any row or any column of a matrix. I have a negative that pops up. I'll take a 3X3 matrix for simplicity.

A=
|a b c|
|d e f|
|g h i|

calculating from the top row we have:
det(A)=a(ei-fh)-b(di-fg)+c(dh-eg) = aei-afh-bdi+bfg+cdh-ceg
calculating from the middle row we have
det(A)=d(bi-ch)-e(ai-cg)+f(ah-bg) = -aei+afh+bdi-bfg-cdh+ceg

The determinant seems to change by a factor of negative 1. This also seems to make sense from row operations. If for example, I swap the first and the second row to yield a matrix B, Then det(B)=-det(A). Yet if I calculate using entries and cofactors corresponding to the second row of B and the first row of A, the cofactor expansion of the two determinants will be identical. Though I'm told that we can start from any row or column, what did I miss? Thanks all.

That's because you are doing the second one incorrectly. The rule is that if you have $a_{ij}$ times its cofactor, you must multiply by $(-1)^{i+j}$. The simplest way to get that factor is to start at the top left and move by horizontally or vertically, alternating "+" and "-".

So for your first calculation, "expanding on the first row", you should have "+, -, +" ($a_{11}$: $(-1)^{1+1}= 1$, $a_{12}$: $(-1)^{1+2}= -1$, $a_{13}$: $(-1)^{1+ 3}= 1$) as you have. But "expanding on the second row", it should be "-, + , -" ($a_{21}$: $(-1)^{2+1}= -1$, $a_{22}$: $(-1)^{2+2}= 1$, $a_{23}$: $(-1)^{2+3}= -1$) which changes the sign.

Last edited by a moderator:
Thank you, perfect, so I was using the minor of each entry which doesn't include the (-1)^(i+j), which is necessary for the cofactor because Cij=(-1)(i+j)Mij. Thank you.

## 1. What are determinants from any row or column?

Determinants from any row or column refer to a mathematical concept used to determine the unique solutions to systems of linear equations. They are calculated using a specific formula and can provide valuable information about the properties of a matrix.

## 2. How do you calculate determinants from any row or column?

To calculate the determinant from any row or column, you must first choose a row or column from the given matrix. Then, you will use a formula that involves multiplying the elements of that row or column by their corresponding cofactors and summing them together. The resulting value is the determinant.

## 3. What is the significance of determinants from any row or column?

Determinants from any row or column have a variety of applications in mathematics, physics, and engineering. They can be used to determine the invertibility of a matrix, the volume of a parallelepiped, and the eigenvalues of a matrix, among other things.

## 4. Can you use determinants from any row or column to solve systems of equations?

Yes, determinants from any row or column can be used to solve systems of linear equations. In fact, they are often used in conjunction with other methods, such as Gaussian elimination, to find the solutions to a system of equations.

## 5. Are all matrices able to have determinants from any row or column calculated?

No, only square matrices (matrices with the same number of rows and columns) can have determinants calculated. The size of the matrix will determine the size of the resulting determinant, as a 2x2 matrix will have a 2x2 determinant, and a 3x3 matrix will have a 3x3 determinant, and so on.

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