“Diagonals” method for 3×3 determinants. Row operations, Levi-Civita, permutations. Using the Casio 991ES. RREF. Video clips.

Diagonals method of calculating 3×3 determinants (it only works for 3×3) – see http://www.purplemath.com/modules/determs2.htm. Or this video:

It works because the “down diagonals” (“down” from left to right) give the even permutations of 123, and the “up diagonals” (“up” from left to right) give the odd permutations.

The Levi-Civita symbol εijk is:

• 1 if ijk is an even permutation of 123, e.g. ε123=1 or
ε312=1
• −1 if ijk is an odd permutation, e.g. ε132=−1 or ε321=−1
• 0 if it’s not a permutation at all, e.g. ε113=0

so the determinant of the 3×3matrix A is

sum over ijk of εijkA1iA2jA3k

Which makes it easier to see why the row operations work:

• Swap two rows, and you multiply the determinant by −1
• Have two rows the same, and the determinant is zero
• Add a multiple of one row to another row, and the determinant stays the same
• Multiply a row by a constant factor m, and you multiply the determinant by m

Here are examples of how you can use those rules to speed the calculation of a determinant:

http://math.stackexchange.com/questions/433870/find-the-determinant-by-using-elementary-row-operations

https://math.dartmouth.edu/archive/m8s00/public_html/handouts/matrices3/node7.html

They both use the fact that you can calculate determinants starting from the first column instead of the first row. (In that case for the diagonals method you write the first and second rows again, below the determinant, instead of the first and second columns again, to the right).

In other words, the determinant of the transpose of a matrix (remember what that is?) is the same as the determinant of the original matrix.

Rules for determinants and inverses

For odd and even permutations see How to invert a matrix using the Casio 991ES calculator

How to calculate a determinant using the Casio 991ES calculator

Reduced row echelon form works for any size of matrix, and is easier than the other methods for 4×4 matrices or bigger, but a little cumbersome for 3×3.

What row operations do to determinants, and how to use row operations to calculate determinants more easily