Mathh 1300: Section 4- 4 Matrices: Basic Operations
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Transcript of Mathh 1300: Section 4- 4 Matrices: Basic Operations
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Math 1300 Finite MathematicsSection 4.4 Matrices: Basic Operations
Jason Aubrey
Department of MathematicsUniversity of Missouri
June 20, 2011
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Two matrices are equal if they have the same size andtheir corresponding elements are equal.
For example,1 3 24 β1 β21 0 β1
=
1 3 24 β1 β21 0 β1
The sum of two matrices of the same size is the matrixwith elements that are the sum of the correspondingelements of the two given matrices.Addition is not defined for matricies of different sizes.
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Two matrices are equal if they have the same size andtheir corresponding elements are equal. For example,1 3 2
4 β1 β21 0 β1
=
1 3 24 β1 β21 0 β1
The sum of two matrices of the same size is the matrixwith elements that are the sum of the correspondingelements of the two given matrices.Addition is not defined for matricies of different sizes.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Two matrices are equal if they have the same size andtheir corresponding elements are equal. For example,1 3 2
4 β1 β21 0 β1
=
1 3 24 β1 β21 0 β1
The sum of two matrices of the same size is the matrixwith elements that are the sum of the correspondingelements of the two given matrices.
Addition is not defined for matricies of different sizes.
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Two matrices are equal if they have the same size andtheir corresponding elements are equal. For example,1 3 2
4 β1 β21 0 β1
=
1 3 24 β1 β21 0 β1
The sum of two matrices of the same size is the matrixwith elements that are the sum of the correspondingelements of the two given matrices.Addition is not defined for matricies of different sizes.
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Evaluate1 1 24 β3 β21 0 β4
+
β2 3 24 β1 β20 0 β4
=
β1 4 48 β4 β41 0 β8
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Evaluate1 1 24 β3 β21 0 β4
+
β2 3 24 β1 β20 0 β4
=
β1 4 48 β4 β41 0 β8
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Evaluate
[5 0 β21 β3 8
]+
β1 70 6β2 8
Undefined!
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Evaluate
[5 0 β21 β3 8
]+
β1 70 6β2 8
Undefined!
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A, B, and C are matrices of the same size, then
A + B = B + A (Commutativity)(A + B) + C = A + (B + C) (Associativity)
A matrix with elements that are all zeros is called a zeromatrix.
For example,[0 00 0
],
0 0 00 0 00 0 0
, etc.
The negative of a matrix M, denoted βM is a matrix withelements that are the negatives of the elements in M. For
example, if M =
[1 β3 22 β4 3
]then βM =
[β1 3 β2β2 4 β3
].
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A, B, and C are matrices of the same size, thenA + B = B + A (Commutativity)
(A + B) + C = A + (B + C) (Associativity)
A matrix with elements that are all zeros is called a zeromatrix.
For example,[0 00 0
],
0 0 00 0 00 0 0
, etc.
The negative of a matrix M, denoted βM is a matrix withelements that are the negatives of the elements in M. For
example, if M =
[1 β3 22 β4 3
]then βM =
[β1 3 β2β2 4 β3
].
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A, B, and C are matrices of the same size, thenA + B = B + A (Commutativity)(A + B) + C = A + (B + C) (Associativity)
A matrix with elements that are all zeros is called a zeromatrix.
For example,[0 00 0
],
0 0 00 0 00 0 0
, etc.
The negative of a matrix M, denoted βM is a matrix withelements that are the negatives of the elements in M. For
example, if M =
[1 β3 22 β4 3
]then βM =
[β1 3 β2β2 4 β3
].
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A, B, and C are matrices of the same size, thenA + B = B + A (Commutativity)(A + B) + C = A + (B + C) (Associativity)
A matrix with elements that are all zeros is called a zeromatrix.
For example,[0 00 0
],
0 0 00 0 00 0 0
, etc.
The negative of a matrix M, denoted βM is a matrix withelements that are the negatives of the elements in M. For
example, if M =
[1 β3 22 β4 3
]then βM =
[β1 3 β2β2 4 β3
].
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A, B, and C are matrices of the same size, thenA + B = B + A (Commutativity)(A + B) + C = A + (B + C) (Associativity)
A matrix with elements that are all zeros is called a zeromatrix.
For example,[0 00 0
],
0 0 00 0 00 0 0
, etc.
The negative of a matrix M, denoted βM is a matrix withelements that are the negatives of the elements in M. For
example, if M =
[1 β3 22 β4 3
]then βM =
[β1 3 β2β2 4 β3
].
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A, B, and C are matrices of the same size, thenA + B = B + A (Commutativity)(A + B) + C = A + (B + C) (Associativity)
A matrix with elements that are all zeros is called a zeromatrix.
For example,[0 00 0
],
0 0 00 0 00 0 0
, etc.
The negative of a matrix M, denoted βM is a matrix withelements that are the negatives of the elements in M.
For
example, if M =
[1 β3 22 β4 3
]then βM =
[β1 3 β2β2 4 β3
].
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A, B, and C are matrices of the same size, thenA + B = B + A (Commutativity)(A + B) + C = A + (B + C) (Associativity)
A matrix with elements that are all zeros is called a zeromatrix.
For example,[0 00 0
],
0 0 00 0 00 0 0
, etc.
The negative of a matrix M, denoted βM is a matrix withelements that are the negatives of the elements in M. For
example, if M =
[1 β3 22 β4 3
]then βM =
[β1 3 β2β2 4 β3
].
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A and B are matrices of the same size, we define subtractionas follows:
A β B = A + (βB)
For example,
[3 1 2β3 3 1
]β[
2 2 β3β1 1 0
]=
[1 β1 5β2 2 1
]
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
If A and B are matrices of the same size, we define subtractionas follows:
A β B = A + (βB)
For example,
[3 1 2β3 3 1
]β[
2 2 β3β1 1 0
]=
[1 β1 5β2 2 1
]
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product of a number k and a matrix M, denoted kM, is amatrix formed by multiplying each element of M by k .
For example,
3
β1 04 3β2 1/3
=
β3 012 9β6 1
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product of a number k and a matrix M, denoted kM, is amatrix formed by multiplying each element of M by k .
For example,
3
β1 04 3β2 1/3
=
β3 012 9β6 1
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Examples: Perform the indicated operations, if possible, giventhat
A =
1 41 33 β7
and B =
0 1β2 24 1
.
3A + B
3 123 99 β21
+
0 1β2 24 1
=
3 131 11
13 β20
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Examples: Perform the indicated operations, if possible, giventhat
A =
1 41 33 β7
and B =
0 1β2 24 1
.
3A + B 3 123 99 β21
+
0 1β2 24 1
=
3 131 11
13 β20
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Examples: Perform the indicated operations, if possible, giventhat
A =
1 41 33 β7
and B =
0 1β2 24 1
.
3A + B 3 123 99 β21
+
0 1β2 24 1
=
3 131 11
13 β20
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
A β B
1 41 33 β7
+
0 β12 β2β4 β1
=
1 33 1β1 β8
βA + 2B
β1 β4β1 β3β3 7
+
0 2β4 48 2
=
β1 β2β5 15 9
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
A β B 1 41 33 β7
+
0 β12 β2β4 β1
=
1 33 1β1 β8
βA + 2B
β1 β4β1 β3β3 7
+
0 2β4 48 2
=
β1 β2β5 15 9
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
A β B 1 41 33 β7
+
0 β12 β2β4 β1
=
1 33 1β1 β8
βA + 2B
β1 β4β1 β3β3 7
+
0 2β4 48 2
=
β1 β2β5 15 9
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
A β B 1 41 33 β7
+
0 β12 β2β4 β1
=
1 33 1β1 β8
βA + 2B β1 β4
β1 β3β3 7
+
0 2β4 48 2
=
β1 β2β5 15 9
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
A β B 1 41 33 β7
+
0 β12 β2β4 β1
=
1 33 1β1 β8
βA + 2B β1 β4
β1 β3β3 7
+
0 2β4 48 2
=
β1 β2β5 15 9
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
DefinitionThe product of a 1Γ n row matrix and an n Γ 1 column matrix isa 1 Γ 1 matrix given by:
[a1 a2 . . . an]
b1b2...
bn
= [a1b1 + a2b2 + Β· Β· Β·+ anbn]
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
DefinitionThe product of a 1Γ n row matrix and an n Γ 1 column matrix isa 1 Γ 1 matrix given by:
[a1 a2 . . . an]
b1b2...
bn
= [a1b1 + a2b2 + Β· Β· Β·+ anbn]
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example:
[1 2 β2
] 021
= [1(0) + 2(2)β 2(1)] = [2]
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example:
[1 2 β2
] 021
= [1(0) + 2(2)β 2(1)]
= [2]
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example:
[1 2 β2
] 021
= [1(0) + 2(2)β 2(1)] = [2]
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Definition (General Matrix Product)If A is an m Γ p matrix and B is a p Γ n matrix, the matrixproduct of A and B, denoted AB, is an m Γ n matrix whoseelements in the i th row and j th column is the real numberobtained from the product of the i th row of A and the j th columnof B. If the number of columns in A does not equal the numberof rows in B, the matrix product AB is not defined.
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 Γ 1 matrix A and a 1 Γ 3 matrix Bhas size
3 Γ 3. However, the product BA has size 1 Γ 1.The product AB of a 2 Γ 4 matrix A and a 4 Γ 3 matrix Bhas size 2 Γ 3. However, the product BA is undefined.
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 Γ 1 matrix A and a 1 Γ 3 matrix Bhas size 3 Γ 3.
However, the product BA has size 1 Γ 1.The product AB of a 2 Γ 4 matrix A and a 4 Γ 3 matrix Bhas size 2 Γ 3. However, the product BA is undefined.
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 Γ 1 matrix A and a 1 Γ 3 matrix Bhas size 3 Γ 3. However, the product BA has size
1 Γ 1.The product AB of a 2 Γ 4 matrix A and a 4 Γ 3 matrix Bhas size 2 Γ 3. However, the product BA is undefined.
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 Γ 1 matrix A and a 1 Γ 3 matrix Bhas size 3 Γ 3. However, the product BA has size 1 Γ 1.
The product AB of a 2 Γ 4 matrix A and a 4 Γ 3 matrix Bhas size 2 Γ 3. However, the product BA is undefined.
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 Γ 1 matrix A and a 1 Γ 3 matrix Bhas size 3 Γ 3. However, the product BA has size 1 Γ 1.The product AB of a 2 Γ 4 matrix A and a 4 Γ 3 matrix Bhas size
2 Γ 3. However, the product BA is undefined.
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 Γ 1 matrix A and a 1 Γ 3 matrix Bhas size 3 Γ 3. However, the product BA has size 1 Γ 1.The product AB of a 2 Γ 4 matrix A and a 4 Γ 3 matrix Bhas size 2 Γ 3.
However, the product BA is undefined.
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 Γ 1 matrix A and a 1 Γ 3 matrix Bhas size 3 Γ 3. However, the product BA has size 1 Γ 1.The product AB of a 2 Γ 4 matrix A and a 4 Γ 3 matrix Bhas size 2 Γ 3. However, the product BA
is undefined.
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
The product AB of a 3 Γ 1 matrix A and a 1 Γ 3 matrix Bhas size 3 Γ 3. However, the product BA has size 1 Γ 1.The product AB of a 2 Γ 4 matrix A and a 4 Γ 3 matrix Bhas size 2 Γ 3. However, the product BA is undefined.
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals
3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0
β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1 73 β 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals
3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0
β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1 73 β 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8)
3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0
β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1 73 β 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8)
3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0
β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13
1 73 β 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2)
3(2) + 1(3)β 2(1)3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0
β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13
1 73 β 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2)
3(2) + 1(3)β 2(1)3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0
β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1
73 β 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)
3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1
73 β 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)
3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1 7
3 β 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)
3(1) + 0 + 0
3(β1) + 0 + 0 3(2) + 1(3) + 0β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1 7
3 β 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)
3(1) + 0 + 0
3(β1) + 0 + 0 3(2) + 1(3) + 0β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1 73
β 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)
3(1) + 0 + 0 3(β1) + 0 + 0
3(2) + 1(3) + 0β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1 73
β 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)
3(1) + 0 + 0 3(β1) + 0 + 0
3(2) + 1(3) + 0β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1 73 β 3
96 0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)
3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0
β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1 73 β 3
96 0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)
3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0
β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1 73 β 3 9
6 0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)
3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0β 2(1) + 0 + 1(8)
β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1 73 β 3 9
6 0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)
3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0β 2(1) + 0 + 1(8)
β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1 73 β 3 96
0 0
Jason Aubrey Math 1300 Finite Mathematics
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Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)
3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2)
β 2(2) + 1(3) + 1(1)
=
β 13 1 73 β 3 96
0 0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)
3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2)
β 2(2) + 1(3) + 1(1)
=
β 13 1 73 β 3 96 0
0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)
3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1 73 β 3 96 0
0
Jason Aubrey Math 1300 Finite Mathematics
university-logo
Addition and SubtractionProduct of a Number k and a Matrix M
The Matrix Product
Example: Find 3 1 β23 1 0β2 1 1
1 β1 20 0 38 β2 1
This equals 3(1) + 0 β 2(8) 3(β1) + 0 β 2(β2) 3(2) + 1(3)β 2(1)
3(1) + 0 + 0 3(β1) + 0 + 0 3(2) + 1(3) + 0β 2(1) + 0 + 1(8) β 2(β1) + 0 + 1(β2) β 2(2) + 1(3) + 1(1)
=
β 13 1 73 β 3 96 0 0
Jason Aubrey Math 1300 Finite Mathematics