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Matrix Inverses and Solving Systems
Warm Up
Lesson Presentation
Lesson Quiz
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Warm UpMultiple the matrices.
1.
Find the determinant.
2. 3. –1 0
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Determine whether a matrix has an inverse.Solve systems of equations using inverse matrices.
Objectives
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multiplicative inverse matrixmatrix equationvariable matrixconstant matrix
Vocabulary
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A matrix can have an inverse only if it is a square matrix. But not all square matrices have inverses. If the product of the square matrix A and the squarematrix A–1 is the identity matrix I, then AA–1 = A–1 A = I, and A–1 is the multiplicative inverse matrix of A, or just the inverse of A.
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The identity matrix I has 1’s on the main diagonal and 0’s everywhere else.
Remember!
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Example 1A: Determining Whether Two Matrices Are Inverses
Determine whether the two given matrices are inverses.
The product is the identity matrix I, so the matrices are inverses.
Example 1B: Determining Whether Two Matrices Are Inverses
Determine whether the two given matrices are inverses.
Neither product is I, so the matrices are not inverses.
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Check It Out! Example 1
Determine whether the given matrices are inverses.
The product is the identity matrix I, so the matrices are inverses.
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If the determinant is 0, is undefined. So a matrix with a determinant of 0
has no inverse. It is called a singular matrix.
Example 2A: Finding the Inverse of a Matrix
Find the inverse of the matrix if it is defined.
First, check that the determinant is nonzero. 4(1) – 2(3) = 4 – 6 = –2. The determinant is –2, so the matrix has an inverse.
The inverse of is
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Example 2A: Finding the Inverse of a Matrix
Find the inverse of the matrix if it is defined.
First, check that the determinant is nonzero. 4(1) – 2(3) = 4 – 6 = –2. The determinant is –2, so the matrix has an inverse.
The inverse of is
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Example 2B: Finding the Inverse of a Matrix
Find the inverse of the matrix if it is defined.
The determinant is, , so B has no inverse.
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Check It Out! Example 2
First, check that the determinant is nonzero.
3(–2) – 3(2) = –6 – 6 = –12
The determinant is –12, so the matrix has an inverse.
Find the inverse of , if it is defined.
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To solve systems of equations with the inverse, you first write the matrix equation AX = B, where A is the coefficient matrix, X is the variable matrix,and B is the constant matrix.
You can use the inverse of a matrix to solve a system of equations. This process is similar to solving an equation such as 5x = 20 by multiplying
each side by , the multiplicative inverse of 5.
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The matrix equation representing is shown.
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To solve AX = B, multiply both sides by the inverse A-1.
A-1AX = A-1B
IX = A-1B
X = A-1B
The product of A-1 and A is I.
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Matrix multiplication is not commutative, so it is important to multiply by the inverse in the sameorder on both sides of the equation. A–1 comes first on each side.
Caution!
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Example 3: Solving Systems Using Inverse Matrices
Write the matrix equation for the system and solve.
Step 1 Set up the matrix equation.
Write: coefficient matrix variable matrix = constant matrix.
A X = B
Step 2 Find the determinant.
The determinant of A is –6 – 25 = –31.
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Example 3 Continued
.
X = A-1 B
Multiply.
Step 3 Find A–1.
The solution is (5, –2).
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Check It Out! Example 3
Step 1 Set up the matrix equation.
A X = B
Step 2 Find the determinant.
The determinant of A is 3 – 2 = 1.
Write the matrix equation for and solve.
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Check It Out! Example 3 Continued
Step 3 Find A-1.
The solution is (3, 1).
X = A-1 B
Multiply.
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Example 4: Problem-Solving Application
Using the encoding matrix ,
decode the message
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List the important information:
• The encoding matrix is E.
• The encoder used M as the message matrix, with letters written as the integers 0 to 26, and then used EM to create the two-row code matrix C.
1 Understand the Problem
The answer will be the words of the message, uncoded.
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2 Make a Plan
Because EM = C, you can use M = E-1C to decode the message into numbers and then convert the numbers to letters.
• Multiply E-1 by C to get M, the message written as numbers.
• Use the letter equivalents for the numbers in order to write the message as words so that you can read it.
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Solve3
Use a calculator to find E-1.
Multiply E-1 by C.
The message in words is “Math is best.”
13 = M, and so on
M A T H _ I
S _ B E S T
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Look Back4
You can verify by multiplying E by M to see that the decoding was correct. If the math had been done incorrectly, getting a different message that made sense would have been very unlikely.
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Check It Out! Example 4
Use the encoding matrix to decode
this message .
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1 Understand the Problem
The answer will be the words of the message, uncoded.
List the important information:
• The encoding matrix is E.
• The encoder used M as the message matrix, with letters written as the integers 0 to 26, and then used EM to create the two-row code matrix C.
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2 Make a Plan
Because EM = C, you can use M = E-1C to decode the message into numbers and then convert the numbers to letters.
• Multiply E-1 by C to get M, the message written as numbers.
• Use the letter equivalents for the numbers in order to write the message as words so that you can read it.
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Solve3
Use a calculator to find E-1.
Multiply E-1 by C.
18 = S, and so onS M A R T Y
_ P A N T S
The message in words is “smarty pants.”
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Lesson Quiz: Part I
yes
1. Determine whether and are
inverses.
2. Find the inverse of , if it exists.
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Lesson Quiz: Part II
Write the matrix equation and solve.
3.
4. Decode using .
"Find the inverse."