1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations 341

DEFINITION Determinant

Let A be a square matrix of size n. The determinant of A is denoted

by IAI or det A.

For n = 1, we define

IAI = det A= det (au) =au.

For n = 2, we define

IAI = det A = det (a^11

a21

And for n = 3, we define

(

a u

IAI = det A = det a21

a31

ai2 ai3)

a22 a23

a32 a33

We could give a general definition for det A or a recursive definition for det A ;

however, since we will only compute det A for A of size n = 1, 2, and 3, the

definitions which we have given will suffice.

We now calculate a few determinants.

1

-1 21 (-1 2)

4

_

3

= det

4

_

3

= (-1)(-3) - (4)(2) = 3-8 = -5,

I

x x21 ( x x

2

)

1 2

x = det

1 2

x = (x)(2x) - (l)(x^2 ) = 2x^2 - x^2 = x^2 ,

-1 0 2

3 1 0

4 -2 3 (

-1 0 2)

= det 3 1 0

4 -2 3

= (-1)(1)(3) + (0)(0)(4) + (2)(3)(-2)

• (-1)(0)(-2) - (0)(3)(3) - (2)(1)(4)

= - 3 + 0 - 12 - 0 - 0 - 8 = -23,