Determinants

th row and the th column column and finding the determinant of the resulting X matrix left behind. We then multiply this quantity by .

For example, suppose

Then the cofactor

and

Finding a Determinant using Cofactors

The cofactors are very useful for finding the determinant.

Pick any row or column, then multiply each element in that row or column by its cofactor, sum these up and you have the determinant.

For example, take the matrix

Arbitrarily, we can choose to expand along the first row. then the determinant of

is

Notice that we said you could expand along any row or column. In the last example, this didn't matter much, but consider the next example.

Example: Find the determinant of

Since I've got three 0s in the second column, I'll simplify my work by expanding along the second column.

If you so desire, you can verify that expanding along any other row or column would produce the same result.