10.4 Day 2 Matrix Algebra 2011 January 19, 2011
10.4 Matrix Algebra DAY 2
Objective: Find the inverse of a matrix. Nov 9­12:38 PM
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10.4 Day 2 Matrix Algebra 2011 January 19, 2011
WARM­UP:
Given the following matrices compute each expression.
[ ]
2 3 2 3 A = 0 ­1 [ ]
B = 3 1 0 2
4 1 ­1 ­2 C =
1. A + C
2. ­3B
3. 2C ­ 3A
4. 5B + C
[ ]
4 3 ­2 0 This is EASY (even if you were absent) so expect to trade papers!!!
Jan 18­10:33 AM
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10.4 Day 2 Matrix Algebra 2011 January 19, 2011
REVIEW:
Given the following matrices, find the product AB .
[ ]
[ ]
2 3 A = 0 ­1 B = 3 1 0 2
­1 ­2 4 1 [ ]
AB = [ ]
AB = Jan 18­10:33 AM
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10.4 Day 2 Matrix Algebra 2011 January 19, 2011
Identity Matrix
Always a square matrix.
[ ]
I2 = 1 0
0 1
[ ]
I3 = Used
inver to find se m
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1 0 0
0 1 0
0 0 1
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Jan 18­11:00 AM
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10.4 Day 2 Matrix Algebra 2011 January 19, 2011
...as long as the dimensions work out the following is true...
Identity Property: IA = A and AI = A
Use the following matrices to demonstrate the property above.
A = [ ]
­1 2 0
3
0 1 [ ]
B = 3 2
4 6
5 2
[ ]
I2 = 1 0
0 1
[ ]
I3 = 1 0 0
0 1 0
0 0 1
[ ] [ ] [ ]
­1 2 0
3
0 1 1 0 0
0 1 0
0 0 1
[ ] [ ] [ ]
3 2
4 6
5 2
3 2
4 6
5 2
1 0
0 1
­1 2 0
[ ] [ ]
3
0 1 1 0
0 1
­1 2 0
0 1 3
[ ]
­1 2 0
0 1 3
Jan 18­12:29 PM
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10.4 Day 2 Matrix Algebra 2011 January 19, 2011
Find the Inverse of a Matrix A­1 is called the inverse of A if the following is true.
­1
­1
AA = I and A A = I
• A is a square matrix.
• A­1 is a square matrix.
• A­1 is read "A inverse."
• A matrix that does have an inverse is called nonsingular.
• A matrix that does NOT have an inverse is called singular.
**If the determinant of A is zero, then A is singular.**
Nov 30­12:50 PM
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10.4 Day 2 Matrix Algebra 2011 January 19, 2011
Procedure for Finding the Inverse of an n by n Matrix
Step 1: Form the matrix [A | In].
Step 2: Transform the matrix [A | In] into reduced row echelon form.
Step 3: Write the inverse of A. The matrix on the left will be the identity matrix and the matrix on the right will be the inverse of A.
Jan 18­12:56 PM
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10.4 Day 2 Matrix Algebra 2011 January 19, 2011
Example #1: Find the inverse of A.
[ ]
3 1
A = 2 1
[ ]
1 ­1
A­1 = ­2 3
Jan 18­2:11 PM
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10.4 Day 2 Matrix Algebra 2011 January 19, 2011
Example #2: Find the inverse of A.
[ ]
4 6
A = 2 3
A­1 = there is no inverse
A is singular.
Jan 18­2:11 PM
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10.4 Day 2 Matrix Algebra 2011 January 19, 2011
Example #3: Find the inverse of A.
[ ]
1 1 0
A = ­1 3 4
0 4 3
[ ]
7
/4 3/4 ­1
A­1 = ­3/4 ­3/4 1
1 1 ­1
Jan 18­2:11 PM
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10.4 Day 2 Matrix Algebra 2011 January 19, 2011
Homework:
page 784 (29 ­ 36, 59, 60)
Jan 18­2:21 PM
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