ch02-05 ppt - Duluth High School
Download
Report
Transcript ch02-05 ppt - Duluth High School
Determinants and
Multiplicative Inverses of
Matrices
Section 2-5
Before finishing this section you should be able to:
• Evaluate determinants
• Find inverses of matrices
• Solve systems of equations by using
inverses of matrices
Remember: Your textbook is your friend! This
presentation is just a supplement to the text.
BEFORE you view this, make sure you read this
section in your textbook and look at all the
great examples that are also worked there for
you.
Determinants
Each square matrix has a determinant. The determinant of
0 2
8 6
is a number denoted by
0 2
8 6
0 2
or det
8 6
The determinant of a 2 x 2
matrix is the difference of the
products of the diagonals.
.
a b
c d
a
d
- b
c
2 = 0(-6) - 8(-2) = 0 + 16 = 16
8 6
Find the value of 0
4 5
If Q
, find the value of det(Q).
3 8
4 5
= (-4)(-8) – (5)(3) = 32 – 15 = 17
3 8
Finding the determinant of a 3x3 matrix
The minor of an element of any nth-order determinant is a
determinant of order (n-1). We take the elements in the
FIRST ROW and find their minors. The minor can be found
by deleting the row and column containing the element.
For instance, find the determinant of
5 3 1
6 4 8
0 3 7
5 3 1
4 8
6 8
6 4
6 4 8 5
3
(1)
3 7
0 7
0 3
0 3 7
This is called expansion by minors.
Notice that this is
SUBTRACTED – the
second term is always
subtracted
Notice that this is
ADDED – the third term
is always added
5 3 1
4 8
6 8
6 4
6 4 8 5
3
(1)
3 7
0 7
0 3
0 3 7
Simplify and expand.
5[4(7)-(-3)(8)] - 3[6(7)-0(8)] + (-1)[6(-3)-0(4)]
= 5[28+24] -3[42-0] + (-1)[-18-0]
= 5[52]-3[42]+(-1)[-18] = 260-126+18 =152
Another Example
Find the value of
.
2 -3 -5
1 2 2
5 3 -1
1 2
2 2
1 2
(5)
2
(3)
5 3
5 1
3 1
= 2(-8) + 3(-11) – 5(-7)
= -14
Identity Matrix
1 0 0
0 1 0
0 0 1
1
0
0
0
0 0 0
1 0 0
0 1 0
0 0 1
The identity matrix for multiplication is a square
matrix whose elements in the main diagonal, from
upper left to lower right, are 1’s, and all other elements
are 0’s.
Any square matrix A multiplied by the identity matrix, I ,
will equal A.
2 3 1 0 2 3
1 8 0 1 1 8
The identity matrix can also be 3 x 3, 4 x 4, 5 x 5,
whatever square dimensions you need for that
particular problem.
Finding the Inverse of a matrix
An inverse matrix, A-1 , is a matrix that, when multiplied by matrix A,
produces the identity matrix.
An inverse matrix of a 2 x 2 matrix can be found by multiplying a
matrix by the reciprocal of the determinant, switching the 2 numbers
of the main diagonal, and changing the signs of the other diagonal,
from top right to bottom left.
For example,
8 9
Find the inverse of the matrix 3 1
First find the determinant
8 9
8( 1) 3(9) 35
3 1
The inverse of the matrix is
1 1 9
35 3 8
9
1
35 35
3
8
35
35
An inverse exists only if the
determinant is not equal to
zero !!!!!!!!!!!!
The reciprocal of the determinant
is multiplied by the altered version
of the original matrix. (the
elements on the main diagonal
switch places, the elements on the
other diagonal change signs).
Another Example
4 2
3 2
Find the inverse of the matrix
.
First, find the determinant of
.
4 2
3 2
4 2
3 2
= 4(2) - 3(2) or 2
The inverse is
1 2 2
2 3 4
1 1
3
2
2
Finding the Inverse of a 3 x 3 matrix
Enter the matrix
2 2 4
3 7 3
5 0 8
We will use our calculator to
find this. It is pretty messy by
hand.
First, enter your matrix.
Go to 2nd, then MATRIX
Hit the x-1 key
Go to MATRIX, Names and hit
ENTER
Scroll over to EDIT and
hit ENTER
To change the decimals to
fractions hit MATH then
ENTER, then ENTER again
Solving a system of equations using matrices
You can use a matrix inverse to solve a matrix equation
in the form AX = B. To solve this equation for X, multiply
each side of the equation by the inverse of A.
AX B
A1 AX A1B
IX A1B
X A1B
Solve the system of equations by using matrix equations.
2x + 3y = -17
Write the system as a
x– y=4
matrix equation.
2 3 x 17
1 1 y 4
To solve the matrix equation, first find the inverse of the
coefficient matrix which is
1 3
5 5
1
2
5
5
Multiply both sides of the matrix equation
by the inverse of the coefficient matrix.
1 3
1 3
5 5 2 3 x 5 5 17
y 1
1
2
1
1
2
4
5
5
5
5
1
1 0 x 5
0 1 y 1
5
3
5 17
2 4
5
x 1
y 5
Example #2: solving a system of equations using matrices
Solve the system of equations by using matrix equations.
3x + 2y = 3
2x – 4y = 2
3 2 x
3
Write the system as a matrix equation.
2 4 y
2
To solve the matrix equation, first find the inverse
of the coefficient matrix.
4 2
1
1 4 2 3
3 2 2 3
16 2 3 2
2 4
Now multiply each side of the matrix equation by the inverse and solve.
1 4 2 3
1 4 2 3 2 x
3 2
16 2
3 2 4 y
16 2
x 1
y
0
The solution is (1, 0).
Real-world example
BANKING A teller at Security Bank received a deposit from a
local retailer containing only twenty-dollar bills and fifty-dollar
bills. He received a total of 70 bills, and the amount of the deposit
was $3200.
How many bills of each value were deposited?
First, let x represent the number of twenty-dollar bills and let y
represent the number of fifty-dollar bills.
So, x + y = 70 since a total of 70 bills were deposited.
Write an equation in standard form that represents the total
amount deposited.
20x + 50y = 3,200
Now solve the system of equations x + y = 70 and 20x + 50y =
3200.
Write the system as a matrix equation and solve.
x + y = 70
20x + 50y = 3200
1 1 x
20 50 y
70
3200
Multiply each side of the equation by the
inverse of the coefficient matrix.
1 50 1 1 1 x
1 20 50 y
30 20
1 50 1 70
30 20 1 3200
x
10
y
60
The solution is (10, 60).
The deposit contained 10 twenty-dollar
bills and 60 fifty-dollar bills.
Helpful Websites
Finding Determinants/Inverses of Matrices:
http://www.purplemath.com/modules/determs.htm
http://www.mathcentre.ac.uk/resources/leaflets/firstaidkits
/5_4.pdf
Solving systems using matrices:
http://math.uww.edu/faculty/mcfarlat/matrix.htm
2-5 self-Check Quiz:
http://www.glencoe.com/sec/math/studytools/cgibin/msgQuiz.php4?isbn=0-07-8608619&chapter=2&lesson=5&quizType=1&headerFile=4&sta
te=