Page 169 - 35Linear Algebra

8

Determinants

Given a square matrix, is there an easy way to know when it is invertible?
Answering this fundamental question is the goal of this chapter.

8.1 The Determinant Formula

The determinant boils down a square matrix to a a single number. That
number determines whether the square matrix is invertible or not. Lets see
how this works for small matrices first.

8.1.1 Simple Examples

For small cases, we already know when a matrix is invertible. If M is a 1 × 1
matrix, then M = (m) ⇒ M −1 = (1/m). Then M is invertible if and only if
m 6= 0.

For M a 2 × 2 matrix, chapter 7 section 7.5 shows that if

m 1 m 2
1 1
M = ,
m 2 1 m 2 2

then
2 1
1 m −m
M −1 = 2 2 .
1
1
2
m m − m m 2 1 −m 2 1 m 1
1
2
1
2
Thus M is invertible if and only if
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