Determinants and Inverses

I.  Homework

II.  Determinants:

Consider row reducing the standard 2x2 matrix

Suppose that a is nonzero.
a b
c d

1/a R1 ->  R1
1 b/a
c d

R2 - cR1  -> R2
1 b/a
0 d - cb/a

Now notice that we cannot make the lower right corner a 1 if

d - cb/a = 0 or

da - cb = 0 or

ad - bc = 0

Definition:  We call ad - bc the determinant of the 2 by 2 matrix
a b
c d

it tells us when it is possible to row reduce the matrix and find a solution to the linear system

Example:  the determinant of the matrix
3 1
5 2

is 3(2) - 1(5) = 6 - 5 = 1

III.  Determinants of Three by Three Matrices

We define the determinant of a triangular matrix

a d e
0 b f
0 0 c

as det = abc

Notice that if we multiply a row by a constant k then the new determinant is k times the old one.

Theorem:  The effect of the the three basic row operations on the determinant are as follows:

1)  Multiplication of a row by a constant multiplies the determinant by that constant.

2)  Switching two rows changes the sign of the determinant

3)  Replacing one row by that row + a multiply of another row has no effect on the determinant.

To find the determinant of a matrix we use the operations to make the matrix triangular and then work backwards.

Example:

2 6 10
2 4 -3
0 4 2

1/2 R1 <-> R1 (Multiplies the determinant by 1/2)

1 3 5
2 4 -3
0 4 2

  R2 - 2R1 -> R2 No effect on the determinant.  Note that we do not need to zero out the upper middle number.  We only need to zero out the bottom left numbers.

1 3 5
0 -2 -13
0 4 2

R3 + 2R2 ->  R3 No effect on the determinant.  Note that we do not need to make the  middle number a 1.

1 3 5
0 -2 -13
0 0 -24

The determinant of this matrix is 48. Since this matrix has 1/2 the determinant of the original matrix, the determinant of the original matrix has determinant 48(2) = 96.

IV.  Inverses

We call the square matrix  I with all 1 down the diagonal and zeros everywhere else the identity matrix.  It has the unique property that if A is a square matrix with the same dimensions then

AI = IA = A

Definition

If A is a square matrix then the inverse A-1  of A is the unique matrix such that

AA-1 = A-1A = I  

Example:  

Let A =
2 5
1 3

then

A-1  =

3 -5
-1 2

If we have time we will verify this.

Theorem:  The inverse of a matrix exists if and only if the determinant is nonzero.

To find the inverse of a matrix, we write a new extended matrix with the identity on the right.  Then we completely row reduce, the resulting matrix on the right will be the inverse matrix.

Example:
2 -1
1 -1

First note that the determinant of this matrix is -2 + 1 = -1, hence the inverse exists.  Now we set the augmented matrix as

2 -1 1 0
1 -1 0 1

R1 <-> R2   
1 -1 0 1
2 -1 1 0

R2 - 2R1 -> R2
1 -1 0 1
0 1 1 -2

R1 + R2  -> R1

1 0 1 -1
0 1 1 -2

Notice that the left hand part is now the identity.  The right hand side is the inverse.  Hence

A-1 =
1 -1
1 -2

V.  Solving Equations Using Matrices

Example:

Suppose we have the system

2x - y = 3

x - y = 4

Then we can write this in matrix form

Ax = b

where

A =
2 -1
1 -1

x =
x
y

and

b =
3
4

We can multiply both sides by A-1:

A-1A x = A-1b

or x = A-1b.

From before,

A-1 =  
1 -1
1 -2

Hence our solution is
-1
-5

or x = -1 and y = 5

VI.  The Easy Way

We will learn how to use the TI 85 calculator to find inverses and determinants