Five-Minute Check (over Lesson 6-2)
Then/Now
New Vocabulary
Key Concept: Invertible Square Linear Systems
Example 1: Solve a 2 × 2 System Using an Inverse Matrix
Example 2: Real-World Example: Solve a 3 × 3 System
Using an Inverse Matrix
Key Concept: Cramer’s Rule
Example 3: Use Cramer’s Rule to Solve a 2 × 2 System
Example 4: Use Cramer’s Rule to Solve a 3 × 3 System
Over Lesson 6-2
Find AB and BA, if possible.
A. BA is not possible; AB =
B. AB is not possible; BA =
C. AB =
; BA =
D. AB is not possible; BA is not possible
Over Lesson 6-2
Find AB and BA, if possible.
A. BA is not possible; AB =
B. AB is not possible; BA =
C. AB =
; BA =
D. AB is not possible; BA is not possible
Over Lesson 6-2
Write the system of equations as a matrix equation,
AX = B. Then use Gauss-Jordan elimination on the
augmented matrix to solve for X.
x1 + 2x2 + 3x3 = –5
2x1 + x2 + x3 = 1
x1 + x2 – x3 = 8
Over Lesson 6-2
A.
; (1, 3, –4)
B.
; (–1, –3, 4)
C.
; (1, 3, –4)
D.
; (–1, –3, 4)
Over Lesson 6-2
A.
; (1, 3, –4)
B.
; (–1, –3, 4)
C.
; (1, 3, –4)
D.
; (–1, –3, 4)
Over Lesson 6-2
For
and
, find AB and BA
and determine whether A and B are inverse
matrices.
A.
B.
C.
D.
Over Lesson 6-2
For
and
, find AB and BA
and determine whether A and B are inverse
matrices.
A.
B.
C.
D.
Over Lesson 6-2
Which of the following represents the determinant
of
A. 0
B. 13
C. 15
D. 17
?
Over Lesson 6-2
Which of the following represents the determinant
of
A. 0
B. 13
C. 15
D. 17
?
You found determinants and inverses of 2 × 2 and
3 × 3 matrices. (Lesson 6-2)
• Solve systems of linear equations using inverse
matrices.
• Solve systems of linear equations using Cramer’s
Rule.
• square system
• Cramer’s Rule
Solve a 2 × 2 System Using an Inverse Matrix
A. Use an inverse matrix to solve the system of
equations, if possible.
2x – y = 1
2x + 3y = 13
Write the system in matrix form AX = B.
AX = B.
Solve a 2 × 2 System Using an Inverse Matrix
Use the formula for the inverse of a 2 × 2 matrix to find
the inverse A–1.
A–1
Formula for the inverse of
a 2 × 2 matrix.
a = 2, b = –1, c = 2, and
d=3
Simplify.
Solve a 2 × 2 System Using an Inverse Matrix
Multiply A–1 by B to solve the system.
X = A–1B
Therefore, the solution of the system is (2, 3).
Answer:
Solve a 2 × 2 System Using an Inverse Matrix
Multiply A–1 by B to solve the system.
X = A–1B
Therefore, the solution of the system is (2, 3).
Answer:
(2, 3)
Solve a 2 × 2 System Using an Inverse Matrix
B. Use an inverse matrix to solve the system of
equations, if possible.
2x + y = 9
x – 3y + 2z = 12
5y – 3z = –11
Write the system in matrix form AX = B.
AX = B
Solve a 2 × 2 System Using an Inverse Matrix
Use a graphing calculator to find A–1.
A–1
Multiply A–1 by B to solve the system.
Solve a 2 × 2 System Using an Inverse Matrix
X = A–1B
Answer:
Solve a 2 × 2 System Using an Inverse Matrix
X = A–1B
Answer:
(5, –1, 2)
Use an inverse matrix to solve the system of
equations, if possible.
2x – 3y = –7
–x – y = 1
A. (–2, 1)
B. (2, –1)
C. (–2, –1)
D. no solution
Use an inverse matrix to solve the system of
equations, if possible.
2x – 3y = –7
–x – y = 1
A. (–2, 1)
B. (2, –1)
C. (–2, –1)
D. no solution
Solve a 3 × 3 System Using an
Inverse Matrix
COINS Marquis has 22 coins that are all nickels,
dimes, and quarters. The value of the coins is
$2.75. He has three fewer dimes than twice the
number of quarters. How many of each type of
coin does Marquis have?
His collection of coins can be represented by
n + d + q = 22
5n + 10d + 25q = 275
d – 2q = –3,
where n, d, and q represent the number of nickels,
dimes, and quarters, respectively.
Write the system in matrix form AX = B.
Solve a 3 × 3 System Using an
Inverse Matrix
Use a graphing calculator to find A–1.
A–1
Solve a 3 × 3 System Using an
Inverse Matrix
Multiply A–1 by B to solve the system.
A–1B
Answer:
Solve a 3 × 3 System Using an
Inverse Matrix
Multiply A–1 by B to solve the system.
A–1B
Answer: 7 nickels, 9 dimes, and 6 quarters
MUSIC Manny has downloaded three types of
music: country, jazz, and rap. He downloaded a
total of 24 songs. Each country song costs $0.75
to download, each jazz song costs $1 to download,
and each rap song costs $1.10 to download. In all
he has spent $23.95 on his downloads. If Manny
has downloaded two more jazz songs than country
songs, how many of each kind of music has he
downloaded?
A. 6 country, 8 jazz, 10 rap
B. 4 country, 6 jazz, 14 rap
C. 5 country, 7 jazz, 12 rap
D. 7 country, 9 jazz, 9 rap
MUSIC Manny has downloaded three types of
music: country, jazz, and rap. He downloaded a
total of 24 songs. Each country song costs $0.75
to download, each jazz song costs $1 to download,
and each rap song costs $1.10 to download. In all
he has spent $23.95 on his downloads. If Manny
has downloaded two more jazz songs than country
songs, how many of each kind of music has he
downloaded?
A. 6 country, 8 jazz, 10 rap
B. 4 country, 6 jazz, 14 rap
C. 5 country, 7 jazz, 12 rap
D. 7 country, 9 jazz, 9 rap
Use Cramer’s Rule to Solve a 2 × 2 System
Use Cramer’s Rule to find the solution to the
system of linear equations, if a unique solution
exists.
4x1 – 5x2 = –49
–3x1 + 2x2 = 28
The coefficient matrix is
determinant of A.
4(2) – (–5)(–3) or –7
. Calculate the
Use Cramer’s Rule to Solve a 2 × 2 System
Because the determinant of A does not equal zero,
you can apply Cramer’s Rule.
So, the solution is x1 = –6 and x2 = 5 or (–6, 5). Check
your answer in the original system.
Answer:
Use Cramer’s Rule to Solve a 2 × 2 System
Because the determinant of A does not equal zero,
you can apply Cramer’s Rule.
So, the solution is x1 = –6 and x2 = 5 or (–6, 5). Check
your answer in the original system.
Answer: (–6, 5)
Use Cramer’s Rule to find the solution of the
system of linear equations, if a unique solution
exists.
–6x + 2y = 28
x – 5y = –14
A. no solution
B. (–4, –2)
C. (4, –2)
D. (–4, 2)
Use Cramer’s Rule to find the solution of the
system of linear equations, if a unique solution
exists.
–6x + 2y = 28
x – 5y = –14
A. no solution
B. (–4, –2)
C. (4, –2)
D. (–4, 2)
Use Cramer’s Rule to Solve a 3 × 3 System
Use Cramer’s Rule to find the solution of the
system of linear equations, if a unique solution
exists.
y + 4z = –1
2x – 2y + z = –18
x – 4z = 7
The coefficient matrix is
the determinant of A.
. Calculate
Use Cramer’s Rule to Solve a 3 × 3 System
Formula for the
determinant of a
3 × 3 matrix
Simplify.
Simplify.
Use Cramer’s Rule to Solve a 3 × 3 System
Because the determinant of A does not equal zero,
you can apply Cramer’s Rule.
Use Cramer’s Rule to Solve a 3 × 3 System
Use Cramer’s Rule to Solve a 3 × 3 System
Therefore, the solution is x = –1, y = 7, and z = –2
or (–1, 7, –2)
Answer:
Use Cramer’s Rule to Solve a 3 × 3 System
Therefore, the solution is x = –1, y = 7, and z = –2
or (–1, 7, –2)
Answer: (–1, 7, –2)
Use Cramer’s Rule to Solve a 3 × 3 System
CHECK Check the solution by substituting back into
the original system.
?
7 + 4(–2) = –1
–1 = –1 
?
2(–1) – 2(7) + –2 = –18
–18 = –18 
?
–1 – 4(–2) = 7
7 = 7
Use Cramer’s Rule to find the solution of the
system of linear equations, if a unique solution
exists.
x – y + 2z = –3
–2x – z = 3
3y + z = 10
A. (2, –3, –1)
B. (–2, 3, 1)
C. (2, 3, 1)
D. no solution
Use Cramer’s Rule to find the solution of the
system of linear equations, if a unique solution
exists.
x – y + 2z = –3
–2x – z = 3
3y + z = 10
A. (2, –3, –1)
B. (–2, 3, 1)
C. (2, 3, 1)
D. no solution