Download Section 9.5: The Algebra of Matrices

Math Analysis Notes
Section 9.5
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Section 9.5: The Algebra of Matrices
Big Idea: Matrix arithmetic needs to be defined carefully, and is different than the arithmetic of real
numbers.
Big Skill: You should be able to multiply matrices, and translate systems of linear equations into matrix
equations.
1. Two matrices A = [aij] and B = [bij] are equal if and only if they have the same dimension (m  n)
(i.e., m rows  n columns) and each corresponding element is equal:
aij = bij] for i = 1, 2, …, m and j = 1, 2, …, n.
2. To add two matrices, add each corresponding element.
3. To subtract two matrices, subtract each corresponding element.
4. To multiply a matrix by a scalar, multiply each element by the scalar.
 b1 
b 
2
5. If  a1 a2 ... an  is a row of matrix A, and   is a column of matrix B, then the inner product is the
 
 
bn 
number a1b1 + a2b2 + … + anbn.
6. The product of an (m  n) matrix A and an (n  k) matrix B is an (m  k) matrix whose elements are
formed by taking the inner product of each row of A with each column of B.
7. Properties of matrix arithmetic:
a. A+ (B + C) = (A + B) + C (associative property of addition)
b. A(BC) = (AB)C (associative property of multiplication)
c. A(B + C) = AB + AC (distributive property)
(B + C)A = BA + CA (distributive property)
d. A + B = B + A (commutative property of addition)
e. Note: matrix multiplication is not commutative for all matrices.
f. Additive identity matrix: [0]
g. Multiplicative identity matrix: The (multiplicative) identity matrix In is the n x n matrix that has
 1 0 0 ... 0 
 0 1 0 ... ...


all zero entries except for 1 on the main diagonal. I n   0 0 ... ... 0 


... ... ... 1 0 
 0 ... 0 0 1 
Practice:
2 3 
 2 7 
2
Let A  
and B  

 . Compute A + B, AB, BA, and A .
5

1
5
1




Math Analysis Notes
Section 9.5
 1 2 3   4 4 1
Compute  2 1 3 7 2 3 .



 1 0 1   3 3 1
 1 2
1 2 3 

Compute 
  3 4 .
4
5
6


 5 6
y  z  0
 x 

Write this system of equations as a matrix equation  x  2 y  5 z  3 .
 3x 
y  z  6

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Math Analysis Notes
Section 9.5
Specialty car manufacturer has the following daily production and profit per car:
January February
Cars Produced Each Day
Model K $1000 $500 
Model K Model R Model W
Model R $2000 $1200   B
Auburn 12 10 0 
Model W $1500 $1000 
Biloxi  4 4 20   A
Chattanooga  8 9 12 
Calculate AB; what does it mean?
What was daily profit in January from Biloxi plant?
What was total daily profit from all three plants in February?
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