1. 9.5 The Algebra of Matrices
1 John 1:9 "If we confess our sins, he is faithful and just to
forgive us our sins and to cleanse us from all unrighteousness."
2. Matrices have many other mathematical applications
apart from solving systems.
In college, math students study Linear Algebra where
many of these applications are explored.
3. Matrices have many other mathematical applications
apart from solving systems.
In college, math students study Linear Algebra where
many of these applications are explored.
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4. Matrices have many other mathematical applications
apart from solving systems.
In college, math students study Linear Algebra where
many of these applications are explored.
Click Me!
This section is an elementary look into Linear Algebra.
5. We can add, subtract, multiply and divide matrices.
6. We can add, subtract, multiply and divide matrices.
Two matrices are equal if their corresponding
elements are equal.
7. We can add, subtract, multiply and divide matrices.
Two matrices are equal if their corresponding
elements are equal.
⎡ 4 3 ⎤ ⎡ 4 9 ⎤
2
⎢ ⎥ ⎢ ⎥
⎢ − 2 = ⎢ 1
0 ⎥ − 0 ⎥
⎢ 6
⎣ ⎥ ⎢ 3
⎦ ⎣ ⎥
⎦
8. Addition
must be of the same dimension (# rows, # columns)
9. Addition
must be of the same dimension (# rows, # columns)
To add, add the corresponding elements.
⎡ −3 5 ⎤ ⎡ −7 −8 ⎤ ⎡ −10 −3 ⎤
⎢ ⎥ + ⎢ ⎥ = ⎢ ⎥
⎣ 2 5 ⎦ ⎣ 0 11 ⎦ ⎣ 2 16 ⎦
10. Addition
must be of the same dimension (# rows, # columns)
To add, add the corresponding elements.
⎡ −3 5 ⎤ ⎡ −7 −8 ⎤ ⎡ −10 −3 ⎤
⎢ ⎥ + ⎢ ⎥ = ⎢ ⎥
⎣ 2 5 ⎦ ⎣ 0 11 ⎦ ⎣ 2 16 ⎦
Subtraction
same dimension needed
11. Addition
must be of the same dimension (# rows, # columns)
To add, add the corresponding elements.
⎡ −3 5 ⎤ ⎡ −7 −8 ⎤ ⎡ −10 −3 ⎤
⎢ ⎥ + ⎢ ⎥ = ⎢ ⎥
⎣ 2 5 ⎦ ⎣ 0 11 ⎦ ⎣ 2 16 ⎦
Subtraction
same dimension needed
To subtract, subtract the corresponding elements.
⎡ −3 5 ⎤ ⎡ −7 −8 ⎤ ⎡ 4 13 ⎤
⎢ ⎥ − ⎢ ⎥ = ⎢ ⎥
⎣ 2 5 ⎦ ⎣ 0 11 ⎦ ⎣ 2 −6 ⎦
12. Addition and Subtraction of Matrices
can be done on your calculator, but it’s usually faster
to do these operations in your head.
13. Addition and Subtraction of Matrices
can be done on your calculator, but it’s usually faster
to do these operations in your head.
Do on calculator ...
⎡ −3 5 ⎤ ⎡ −7 −8 ⎤
⎢ ⎥ + ⎢ ⎥
⎣ 2 5 ⎦ ⎣ 0 11 ⎦
⎡ −3 5 ⎤ ⎡ −7 −8 ⎤
⎢ ⎥ − ⎢ ⎥
⎣ 2 5 ⎦ ⎣ 0 11 ⎦
15. Scaler Multiplication
a scaler is a constant
Scaler Multiplication of a matrix uses the distributive
property ... multiply each element by the scaler.
⎡ a11 a12 ⎤ ⎡ c ⋅ a11 c ⋅ a12 ⎤
c ⎢ ⎥ = ⎢ ⎥
⎢ a21 a22
⎣ ⎥ ⎢ c ⋅ a21 c ⋅ a22
⎦ ⎣ ⎥
⎦
17. Algebraic Properties of Matrices
A, B, C are m x n matrices
c, d are scalers
A+ B= B+ A Commutative for Addition
18. Algebraic Properties of Matrices
A, B, C are m x n matrices
c, d are scalers
A+ B= B+ A Commutative for Addition
( A + B) + C = A + ( B + C ) Associative for Addition
19. Algebraic Properties of Matrices
A, B, C are m x n matrices
c, d are scalers
A+ B= B+ A Commutative for Addition
( A + B) + C = A + ( B + C ) Associative for Addition
c ( dA ) = ( cd ) A Associative for Multiplication
20. Algebraic Properties of Matrices
A, B, C are m x n matrices
c, d are scalers
A+ B= B+ A Commutative for Addition
( A + B) + C = A + ( B + C ) Associative for Addition
c ( dA ) = ( cd ) A Associative for Multiplication
( c + d ) A = cA + dA Distributive
21. Algebraic Properties of Matrices
A, B, C are m x n matrices
c, d are scalers
A+ B= B+ A Commutative for Addition
( A + B) + C = A + ( B + C ) Associative for Addition
c ( dA ) = ( cd ) A Associative for Multiplication
( c + d ) A = cA + dA Distributive
c ( A + B ) = cA + cB Distributive
22. ⎡ 4 −1 ⎤ ⎡ −13 1 ⎤
A = ⎢ ⎥ B = ⎢ ⎥
⎣ 3 2 ⎦ ⎣ −3 −2 ⎦
Find matrix X such that
3X − 4A = B
23. ⎡ 4 −1 ⎤ ⎡ −13 1 ⎤
A = ⎢ ⎥ B = ⎢ ⎥
⎣ 3 2 ⎦ ⎣ −3 −2 ⎦
Find matrix X such that
3X − 4A = B
3X = B + 4A
24. ⎡ 4 −1 ⎤ ⎡ −13 1 ⎤
A = ⎢ ⎥ B = ⎢ ⎥
⎣ 3 2 ⎦ ⎣ −3 −2 ⎦
Find matrix X such that
3X − 4A = B
3X = B + 4A
⎡ −13 1 ⎤ ⎡ 16 −4 ⎤
3X = ⎢ ⎥ + ⎢ ⎥
⎣ −3 −2 ⎦ ⎣ 12 8 ⎦
