Determinants & Inverse Matrices
Determinants & Inverse Matrices
The
of the 2 2 matrix
determinant
ab
cd
is the number
.
ad cb
The above sentence is abbreviated as
det a b = ad cb
cd
Example.
det
4 1
2 3
= 4(
3)
1( 2) = 12 + 2 = 10
The determinant of a 3 3 matrix can be found using the formula
0
1
ab c
det @
A = det e f
det d f + det d e
def a
b
c
hi
gi
gh
gh i
Example.
0 21
det @0 3 10
0 2 1
1
A = 2 det
3 0
2 1
(
1) det
0 1
2 1
+ 0 det
0 1
3 0
= 2[3 ? 1 0( 2)] + [0 ? 1 1( 2)] + 0
=2?3+2
=8
*************
286
Determinants and inverses
A matrix has an inverse exactly when its determinant is not equal to 0.
*************
22 inverses
Suppose that the determinant of the 2 2 matrix
ab
cd
does not equal 0. Then the matrix has an inverse, and it can be found using the formula
1
ab =
1
d
b
cd
det a b
ca
cd
Notice that in the above formula we are allowed to divide by the determinant since we are assuming that it's not 0.
Example. To find
35 1 12
first check that
det
3 1
5 2
=3?2
1?5=1
Then
35 12
1=1 1
2 1
5 3
=
2 1
5 3
*************
287
33 inverses
There is a way to find an inverse of a 3 3 matrix ? or for that matter, an matrix ? whose determinant is not 0, but it isn't quite as simple as
nn finding the inverse of a 2 2 matrix. You can learn how to do it if you take a linear algebra course. You could also find websites that will invert matrices for you, and some calculators can find the inverses of matrices as long as the matrices are not too large.
*************
288
Exercises
For #1-6, compute the determinant of the given matrix.
1.)
21
11
2.)
1
01
3.)
42
21
4.)
0
1
3 00
@1p07 1 0A 2 26
5.)
0
1
111
@2 2 2A
333
6.)
0
1
121
@2 1 2A
134
7.) Which of the six matrices from the previous problems have inverses?
Find the inverses of the matrices below.
8) .
10 51
9) .
35 23
289
10 ) .
42 13
I
/
a
Match the functions with their graphs. (
11.) ( ) = 2 fx x
13.) ( ) = x px
2
x
if 2 ( 1 0);
x
,
if
x
2
[0 ,
1).
I 12.) ( ) = gx x
J7, i (
2
14.)
() qx
=I
x
r
if 2 ( x
1 ,
02);
if 2 [0 1).
xx,
1.
A.)
B.)
I / II I
\
4--'
a
--2
I I
C.)
290
II // D.)
i
7,
27
285
\
I
4--'
290
i i --2
I
4--'
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- lecture 18 university of richmond
- the determinant a means to calculate volume
- section 2 3 properties of determinants
- chapter 8 matrices and determinants
- determinants and eigenvalues
- chapter 4 determinants
- determinants of 3×3 matrices date period
- positive definite matrix
- 5 3 determinants and cramer s rule university of utah
- new method to compute the determinant of a 3x3 matrix
Related searches
- matrices calculator solver
- matrices calculator with steps
- solving systems using matrices calculator
- matrices and determinants calculator
- determinants of matrices calculator
- matrices and determinants pdf
- inverse matrices pdf
- biological determinants of health
- properties of determinants 3x3
- determinants of 3x3 matrices answers
- converting matrices into augmented matrices calculator
- determinants of service quality pdf