"All limits, especially national ones, are contrary to the nature of mathematics… Mathematics knows no races… For mathematics the whole cultural world is a single country" – David Hilbert, the foremost mathematician of the 20th century, speaking as the Nazis neared power in his native Germany.

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

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 ε_{ijk} is:

• 1 if ijk is an even permutation of 123, e.g. ε_{123}=1 or
ε_{312}=1
• −1 if ijk is an odd permutation, e.g. ε_{132}=−1 or ε_{321}=−1
• 0 if it’s not a permutation at all, e.g. ε_{113}=0

so the determinant of the 3×3matrix A is

sum over ijk of ε_{ijk}A_{1i}A_{2j}A_{3k}

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:

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.

How to invert a matrix using the Casio 991ES calculatorHow to calculate a determinant using the Casio 991ES calculatorReduced 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