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Lab A Lab B
Computers 15 27
Computer Tables 16 34
Chairs 16 34

Converting the data to a matrix, we have

C 2013 = [ 15 16 16 27 34 34 ]

To calculate how much computer equipment will be needed, we multiply all entries in matrix C by 0.15.

( 0.15 ) C 2013 = [ ( 0.15 ) 15 ( 0.15 ) 16 ( 0.15 ) 16 ( 0.15 ) 27 ( 0.15 ) 34 ( 0.15 ) 34 ] = [ 2.25 2.4 2.4 4.05 5.1 5.1 ]

We must round up to the next integer, so the amount of new equipment needed is

[ 3 3 3 5 6 6 ]

Adding the two matrices as shown below, we see the new inventory amounts.

[ 15 16 16 27 34 34 ] + [ 3 3 3 5 6 6 ] = [ 18 19 19 32 40 40 ]

This means

C 2014 = [ 18 19 19 32 40 40 ]

Thus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables, and 40 chairs.

Scalar multiplication

Scalar multiplication involves finding the product of a constant by each entry in the matrix. Given

A = [ a 11 a 12 a 21 a 22 ]

the scalar multiple c A is

c A = c [ a 11 a 12 a 21 a 22 ]      = [ c a 11 c a 12 c a 21 c a 22 ]

Scalar multiplication is distributive. For the matrices A , B , and C with scalars a and b ,

a ( A + B ) = a A + a B ( a + b ) A = a A + b A

Multiplying the matrix by a scalar

Multiply matrix A by the scalar 3.

A = [ 8 1 5 4 ]

Multiply each entry in A by the scalar 3.

3 A = 3 [ 8 1 5 4 ] =   [ 3 8 3 1 3 5 3 4 ] =   [ 24 3 15 12 ]
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Given matrix B , find −2 B where

B = [ 4 1 3 2 ]

−2 B = [ −8 −2 −6 −4 ]

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Finding the sum of scalar multiples

Find the sum 3 A + 2 B .

A = [ 1 −2 0 0 −1 2 4 3 −6 ]  and  B = [ −1 2 1 0 −3 2 0 1 −4 ]

First, find 3 A , then 2 B .

3 A = [ 3 1 3 ( −2 ) 3 0 3 0 3 ( −1 ) 3 2 3 4 3 3 3 ( −6 ) ] = [ 3 −6 0 0 −3 6 12 9 −18 ]
2 B = [ 2 ( −1 ) 2 2 2 1 2 0 2 ( −3 ) 2 2 2 0 2 1 2 ( −4 ) ] = [ −2 4 2 0 −6 4 0 2 −8 ]

Now, add 3 A + 2 B .

3 A + 2 B = [ 3 −6 0 0 −3 6 12 9 −18 ] + [ −2 4 2 0 −6 4 0 2 −8 ]               = [ 3 2 −6 + 4 0 + 2 0 + 0 −3 6 6 + 4 12 + 0 9 + 2 −18 −8 ]               = [ 1 −2 2 0 −9 10 12 11 26 ]
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Finding the product of two matrices

In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If A is an   m   ×   r   matrix and B is an   r   ×   n   matrix, then the product matrix A B is an   m   ×   n   matrix. For example, the product A B is possible because the number of columns in A is the same as the number of rows in B . If the inner dimensions do not match, the product is not defined.

We multiply entries of A with entries of B according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.

To obtain the entries in row i of A B , we multiply the entries in row i of A by column j in B and add. For example, given matrices A and B , where the dimensions of A are 2   ×   3 and the dimensions of B are 3   ×   3 , the product of A B will be a 2   ×   3 matrix.

A = [ a 11 a 12 a 13 a 21 a 22 a 23 ]  and  B = [ b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 ]

Multiply and add as follows to obtain the first entry of the product matrix A B .

  1. To obtain the entry in row 1, column 1 of A B , multiply the first row in A by the first column in B , and add.
    [ a 11 a 12 a 13 ] [ b 11 b 21 b 31 ] = a 11 b 11 + a 12 b 21 + a 13 b 31
  2. To obtain the entry in row 1, column 2 of A B , multiply the first row of A by the second column in B , and add.
    [ a 11 a 12 a 13 ] [ b 12 b 22 b 32 ] = a 11 b 12 + a 12 b 22 + a 13 b 32
  3. To obtain the entry in row 1, column 3 of A B , multiply the first row of A by the third column in B , and add.
    [ a 11 a 12 a 13 ] [ b 13 b 23 b 33 ] = a 11 b 13 + a 12 b 23 + a 13 b 33

Questions & Answers

Complementary angles
Idrissa Reply
Commplementary angles
Idrissa Reply
Complementary angles
Idrissa
hello
Sherica
im all ears I need to learn
Sherica
Complementary angles
Idrissa
yes
Sherica
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
Practice Key Terms 5

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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