Elementary Row and Column Operations

Recall the elementary row operations of matrices. We can do something similar with determinants to make calculating them easier by reducing certain values to 0.

Theorem

Let

  1. If has a row (or column) of 0, then
  2. If is obtained from by swapping two distinct rows (or two distinct columns), then
  3. If is obtained from by adding a multiple of one row to another row (or a multiple of one column to another column), then
  4. If is obtained from by multiplying a row (or a column) by , then
Important

Do not perform elementary row operations and elementary column operations at the same time. Do them in two steps.

Example

Find if .

Triangular Matrices

Upper and Lower Triangular Matrices

Definition

Let . is called upper triangular if every entry below the main diagonal is zero. is called lower triangular if every entry above the main diagonal is zero.

Example

Upper triangular matrices:

Lower triangular matrices:

Determinant of Triangular Matrix Theorem

Theorem

If is a triangular matrix, then:

(see Summation and Product Notation)