Matrix determinant lemma

The matrix determinant lemma is a way to easily figure out the determinant of a certain type of matrix. A determinant is a number that describes the size and shape of a matrix. The matrix determinant lemma tells you a special way to find the determinant of a matrix that has an extra row or column added on the end.

To use the matrix determinant lemma, you need to find the determinant of an already existing matrix (called the principal matrix) and then you multiply it by the number that is in the extra row or column (called the stride). Finally, you add or subtract the product of the stride and the determinant of the principal matrix depending on if the stride is positive or negative.

For example, let's say the principal matrix is a 3x3 matrix and the stride is 5. First you can find the determinant of the principal matrix by multiplying the numbers that are diagonally across from one another, adding those products together and then subtracting the products of the numbers that are in a line vertically or horizontally. For example if your principal matrix is:

2 3 4
6 5 1
9 7 8

then you would multiply 2 and 8 (2 x 8 = 16), 3 and 7 (3 x 7 = 21), and 4 and 6 (4 x 6 = 24). Then you add the products together (16 + 21 + 24 = 61). Finally, you subtract the products of the numbers in a line horizontally or vertically, for our example that would be 6 x 4 (6 x 4 = 24) and 9 x 3 (9 x 3 = 27), so 24 - 27 = -3. So our determinant for the principal matrix is 61 - 3 = 58.

Finally, multiply the stride and the determinant together, 5 x 58 = 290. Then, depending on if the stride was positive or negative, you either add or subtract the product from the determinant of the principal matrix. So if our stride was positive the determinant of the matrix after the extra row or column is added is 58 + 290 = 348. And if our stride was negative the determinant of the matrix after the extra row or column is added is 58 - 290 = -232.

So that is the matrix determinant lemma in a nutshell.