Downhill product

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Downhill product Uphill product.
Copy columns 1 2 to the outside.
C1 C2
Downhill products Uphill products.
2
C1 C2
(2)(5)(-9)
(-1)(1)(0)
(3)(-2)(6)
(0)(5)(3)
(6)(1)(2)
(-9)(-2)(-1)
-90
0
(-36)
0
12
(-18)
-120
C1 C2
(2)(10)(0)
(3)(2)(6)
(0)(10)(3)
(6)(10)(1)
(4)(2)(2)
(1)(10)(4)
0
36
0
60
16
40
0
3
Uses determinants to solve systems of equations.
Matrix Dy is the coefficient matrix with the
results in the y column..
Matrix D is the coefficient matrix.
Matrix Dx is the coefficient matrix with the
results in the x column.
4
Set up all the determinants to solve the 3 by 3
system by Cramers Rule.
Find the coefficient matrix first.
Find Matrix Dx , Dy , and Dz for the numerators.
5
Solve the system by Cramers Rule.
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Solve the system by Cramers Rule.
Remember to always find Matrix D 1st. If this
matrix determinant is equal to zero, then there
is no way to determine the answer. It is either
NO SOLUTION, or INFINITE SOLUTION. We cant tell.
7
Rows by Columns.
21 means Row 2 Column 1. If there are 2 digit
rows or columns, a comma is used.
Equivalent Matrices must have the same dimensions
and all corresponding entries must be the same.
Find the values of x, y, and z.
x 4
y - 8
z 2
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(No Transcript)
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Zero Matrix is when every entry is zero.
Distributive Property with a Matrix..
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Same
2 x 3
3 x 2
Same
The product is a 2 x 2 Matrix. Make an empty
matrix.
2 ( 4 ) 1 ( 8 ) 3 ( 5 )
2 ( -7 ) 1 ( 6 ) 3 ( 9 )
1 ( 4 ) -1 ( 8 ) 0 ( 5 )
1 ( -7 ) -1 ( 6 ) 0 ( 9 )
1. Row 1 of A times Column 1 of B
2. Row 1 of A times Column 2 of B
3. Row 2 of A times Column 1 of B
4. Row 2 of A times Column 2 of B
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The product is a 3 x 3 Matrix. Make an empty
matrix.
Find BA
4 ( 2 ) -7 ( 1 )
4 ( 1 ) -7 ( -1 )
4 ( 3 ) -7 ( 0 )
8 ( 2 ) 6 ( 1 )
8 ( 1 ) 6 ( -1 )
8 ( 3 ) 6 ( 0 )
5 ( 2 ) 9 ( 1 )
5 ( 1 ) 9 ( -1 )
5 ( 3 ) 9 ( 0 )
2 x 3
3 x 2
Same
1. R1 of B times C1 of A
2. R1 of B times C2 of A
3. R1 of B times C3 of A
4. R2 of B times C1 of A
5. R2 of B times C2 of A
6. R2 of B times C3 of A
7. R3 of B times C1 of A
8. R3 of B times C2 of A
9. R3 of B times C3 of A
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The Identity Matrix is a square matrix, n by
n matrix, which has 1s on the downhill diagonal
and 0s for all the other entries.
Identity Matrix
Inverse Matrix.
Show that A and B are inverses.

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One way is to create a variable matrix and
multiply it to the matrix you are asked to find
the inverse and set the product equal to the
Identity Matrix. This creates systems of
equations to solve.
How to find the inverse of a matrix?
2. R1 C2
1. R1 C1
4. R2 C2
3. R2 C1
Subtract the equations.

Back substitute.
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Here is another way to find the inverse of a
matrix.
We can also make an Augmented Matrix, page 5 of
the notes, and use the Gauss-Jordan Method to
covert the matrix.
Gauss-Jordan Method
2
-2
-0
-2
R1 R2 R1
-2R1 R2 R2
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Solving systems of equations using the inverse
matrix method.
This technique is all calculator. We need to
create a Linear Matrix Equation.
x
6
-1
1
1
y
1
-2
3
-5
z
3
1
-2
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We need to create a Coefficient Matrix
Variable Matrix
Result Matrix.
Store Matrix A and B in calculator.
The Linear Matrix Equation.
Multiply both side by the inverse of Matrix A.
Find the inverse of Matrix A.
Determinant of A is the denominator in the
inverse.
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3 ways to solve systems of equations

• Gauss-Jordan Elimination Method, rref(
• Cramers Rule,
  determinants
• Inverse Matrix Method,
  inverse matrices

This is the best technique! The other two
techniques cant determine when a system has No
Solution or Infinite Solutions!