Determinants

The determinant of an upper triangular matrix is the product of its diagonal entries.

We can find the determinant of a matrix through row reduction. We have to keep track of a multiplier for the determinant of the upper triangular matrix:

• each time we swap a row, multiply the determinant by -1

• each time we multiply a row by a scalar, multiply the determinant by that scalar

• adding a multiple of one row to another doesn't change the determinant.

Determinants and Cramer's rule

For 2x2 matrices, , the determinant is $ad-bc$.

For a system of two equations with two unknowns:

$$a_{1}x+b_{1}y=q_1,a_{2}x+b_{2}y=q_2$$

The unique solution can be given via Cramer's rule as:

3x3 determinant

For the 3x3 matrix:

the determinant is: