Solving Systems with Inverse Matrices
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Transcript Solving Systems with Inverse Matrices
Sec. 7.3c
Let A be the coefficient matrix of a system of n linear
equations in n variables given by AX = B, where X is the
n x 1 matrix of variables and B is the n x 1 matrix of
numbers of the right-hand side of the equations. If A–1
exists, then the system of equations has the unique solution
X=A
–1
B
Write the system of equations as a matrix equation AX = B,
with A as the coefficient matrix of the system.
x 3y z 9
2x 4z 1
8 x y z 5
1 3 1 x 9
AX = B: 2 0 4 y 1
8 1 1 z 5
Write the matrix equation as a system of equations
1 2 3 1 x 2 x 2 y 3 z w 2
0 0 2 8 y 3
2z 8w 3
9 0 1 5 z 9
9x z 5w 9
1 1 6 3 w 2 x y 6 z 3w 2
Solve the given system using inverse matrices
3x 2 y 0
0
x
3 2
X B
x y 5 A
5
y
1 1
To solve for X, apply the
inverse of A to both sides
of the matrix equation:
10
X A B
15
1
3 x 2 y
A X
B
x y
Solution:
(x, y) = (10, 15)
Solve the given system using inverse matrices
3x 3 y 6 z 20
x 3 y 10 z 40
x 3 y 5 z 30
Find
1
XA B
Solution:
(x, y, z) = (18, 118/3, 14)
3 3 6
x
20
A 1 3 10 X y B 40
30
1 3 5
z
Solve the given system using inverse matrices
x 4 y 2z 0
2x y z 6
3x 3 y 5 z 13
Find
1
XA B
Solution:
(x, y, z) = (3, –1/2, 1/2)
1 4 2
x
0
A 2 1 1 X y B 6
3 3 5
z
13
Solve the given system using inverse matrices
2x y 2z 8
3x 2 y z w 10
2 x 3w y 1
4 x 3 y 2 z 5w 39
Find
1
XA B
Solution:
(x, y, z, w) =
(4, –2, 1, –3)
2 1 2 0
x
8
3 2 1 1
y
10
X B
A
2 1 0 3
z
1
4 3 2 5
w
39
Use a method of your choice to solve the given system.
x yz 6
x y 2 z 2
Augmented Matrix:
1 1 1 6
1 1 2 2
1
0
1.5
2
RREF:
0 1 0.5 4
Solution:
(x, y, z) = (2 – 1.5z, –4 – 0.5z, z)
Fitting a parabola to three points. Determine a, b, and c so
that the points (–1, 5), (2, –1), and (3, 13) are on the graph of
f x ax bx c
2
How about a diagram to start???
We need f(–1) = 5, f(2) = –1, and f(3) = 13:
f 1 a b c 5
f 2 4a 2b c 1
f 3 9a 3b c 13
Now, simply solve
this system!!!
(a, b, c) = (4, –6, –5)
f x 4x 6x 5
2
Double-check with a graph?
Mixing Solutions. Aileen’s Drugstore needs to prepare a 60-L
mixture that is 40% acid using three concentrations of acid. The
first concentration is 15% acid, the second is 35% acid, and the
third is 55% acid. Because of the amounts of acid solution on
hand, they need to use twice as much of the 35% solution as
the 55% solution. How much of each solution should they use?
x = liters of 15% solution
y = liters of 35% solution
z = liters of 55% solution
x y z 60
y 2z 0
0.15x 0.35 y 0.55z 0.40 60
Solve the system!!!
Need 3.75 L of 15% acid, 37.5 L of 35% acid, and
18.75 L of 55% acid to make 60 L of 40% acid solution.
Manufacturing. Stewart’s metals has three silver alloys on hand.
One is 22% silver, another is 30% silver, and the third is 42%
silver. How many grams of each alloy is required to produce 80
grams of a new alloy that is 34% silver if the amount of 30% alloy
used is twice the amount of 22% alloy used?
x = amount of 22% alloy
y = amount of 30% alloy
z = amount of 42% alloy
x y z 80
0.22 x 0.30 y 0.42 z 27.2
2x y 0
Solve the system!!!
Need approximately 14.545g of the 22% alloy,
29.091g of the 30% alloy, and 36.364g of the 42% alloy
to make 80g of the 34% alloy.
Vacation Money. Heather has saved $177 to take with her on
the family vacation. She has 51 bills consisting of $1, $5, and
$10 bills. If the number of $5 bills is three times the number of
$10 bills, find how many of each bill she has.
x = number of $1 bills
y = number of $5 bills
z = number of $10 bills
x y z 51
x 5 y 10 z 177
y 3z 0
Solve the system!!!
Heather has 27 one-dollar bills, 18 fives, and 6 tens.