Linear Algebra Matrix-Cofactor Expansion?

Question

I have a 4x4 matrix which has been reduced by elementary operations and started the cofactor expansion to get 2 3x3 matrixs (the other 2 3x3 are multiplied by 0).

My question is can I continue to use elementary operations to reduce the 3x3 matrices and then apply the cofactor expansion again to receive 2 2x2s?

i.e, as per where I'm at with my problem, does | D | = a11 *A11 + a31*A31= a11*b11*B11+a11*b31*B31?

I haven't been able to find any information online anywhere so if you have a source, please link it.

Thank you.

Answer 1

"My question is can I continue to use elementary operations to reduce the 3x3 matrices and then apply the cofactor expansion again to receive 2 2x2s?"

You can use SOME elementary row operations without changing the determinant - specifically, you can use the operation of subtracting one row from another row without changing the determinant at all. If you want to prove this to yourself, first verify that it doesn't change the determinant of a 2×2 matrix. Then proceed by induction - suppose that the determinant of an n×n matrix is unchanged. Note that you can expand an (n+1)×(n+1) matrix along a row which is not the row you subtracted, or the row you subtracted it from. The result of this expansion will be n+1 minors multiplied by the same cofactors as the original matrix, and each minor will be an n×n matrix which is the same as the n×n matrix that you would have gotten by expanding the original matrix, but with a multiple of one row added to another. However, we have already supposed that that doesn't change the determinant of n×n matrices - therefore the cofactors and the determinants of the minors are all the same as the original matrix, meaning that an (n+1)×(n+1) matrix with a multiple of one row added to another has the same determinant as the original (n+1)×(n+1) matrix. Since row addition doesn't change the determinant of 2×2 matrices, it therefore doesn't change the determinant of 3×3 matrices, and from that we deduce that it doesn't change the determinants of 4×4, 5×5, 6×6, or in general, any n×n matrix. Q.E.D.

The other elementary row operations do have effects on the determinant - multiplying a row by a scalar will multiply the determinant by that scalar, and swapping two rows will cause the determinant to be multiplied by -1. But as long as you stick to just row addition, you won't do a thing to the determinant. Actually, this is a huge timesaver - instead of using cofactor expansion, I usually reduce the matrix to an upper-triangular form first and then find the determinant by multiplying the numbers on the main diagonal (you may easily verify that this gives the same result as cofactor expansion for matrices in upper-triangular form). This is much faster than using cofactor expansion on large matrices - the row-reduction takes only O(n³) time whereas naive cofactor expansion on a zero-poor matrix takes O(n!) - I'm sure you can see the advantage.

- Pascal

Answer 2

yes
you are correct
you can continue to use elementary operations to reduce the 3x3 matrices and then apply the cofactor expansion again to receive 2 2x2s

Answer 3

My answer is maybe.