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Using a calculator to perform matrix operations

Find A B C given

A = [ −15 25 32 41 −7 −28 10 34 −2 ] , B = [ 45 21 −37 −24 52 19 6 −48 −31 ] , and  C = [ −100 −89 −98 25 −56 74 −67 42 −75 ] .

On the matrix page of the calculator, we enter matrix A above as the matrix variable [ A ] , matrix B above as the matrix variable [ B ] , and matrix C above as the matrix variable [ C ] .

On the home screen of the calculator, we type in the problem and call up each matrix variable as needed.

[ A ] × [ B ] [ C ]

The calculator gives us the following matrix.

[ 983 462 136 1 , 820 1 , 897 856 311 2 , 032 413 ]
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Access these online resources for additional instruction and practice with matrices and matrix operations.

Key concepts

  • A matrix is a rectangular array of numbers. Entries are arranged in rows and columns.
  • The dimensions of a matrix refer to the number of rows and the number of columns. A 3 × 2 matrix has three rows and two columns. See [link] .
  • We add and subtract matrices of equal dimensions by adding and subtracting corresponding entries of each matrix. See [link] , [link] , [link] , and [link] .
  • Scalar multiplication involves multiplying each entry in a matrix by a constant. See [link] .
  • Scalar multiplication is often required before addition or subtraction can occur. See [link] .
  • Multiplying matrices is possible when inner dimensions are the same—the number of columns in the first matrix must match the number of rows in the second.
  • The product of two matrices, A and B , is obtained by multiplying each entry in row 1 of A by each entry in column 1 of B ; then multiply each entry of row 1 of A by each entry in columns 2 of B , and so on. See [link] and [link] .
  • Many real-world problems can often be solved using matrices. See [link] .
  • We can use a calculator to perform matrix operations after saving each matrix as a matrix variable. See [link] .

Section exercises

Verbal

Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.

No, they must have the same dimensions. An example would include two matrices of different dimensions. One cannot add the following two matrices because the first is a 2 × 2 matrix and the second is a 2 × 3 matrix. [ 1 2 3 4 ] + [ 6 5 4 3 2 1 ] has no sum.

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Can we multiply any column matrix by any row matrix? Explain why or why not.

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Can both the products A B and B A be defined? If so, explain how; if not, explain why.

Yes, if the dimensions of A are m × n and the dimensions of B are n × m , both products will be defined.

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Can any two matrices of the same size be multiplied? If so, explain why, and if not, explain why not and give an example of two matrices of the same size that cannot be multiplied together.

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Does matrix multiplication commute? That is, does A B = B A ? If so, prove why it does. If not, explain why it does not.

Not necessarily. To find A B , we multiply the first row of A by the first column of B to get the first entry of A B . To find B A , we multiply the first row of B by the first column of A to get the first entry of B A . Thus, if those are unequal, then the matrix multiplication does not commute.

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Algebraic

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.

A = [ 1 3 0 7 ] , B = [ 2 14 22 6 ] , C = [ 1 5 8 92 12 6 ] , D = [ 10 14 7 2 5 61 ] , E = [ 6 12 14 5 ] , F = [ 0 9 78 17 15 4 ]

C + D

[ 11 19 15 94 17 67 ]

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D B

Undidentified; dimensions do not match

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For the following exercises, use the matrices below to perform scalar multiplication.

A = [ 4 6 13 12 ] , B = [ 3 9 21 12 0 64 ] , C = [ 16 3 7 18 90 5 3 29 ] , D = [ 18 12 13 8 14 6 7 4 21 ]

3 B

[ 9 27 63 36 0 192 ]

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−4 C

[ −64 −12 −28 −72 −360 −20 −12 −116 ]

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100 D

[ 1 , 800 1 , 200 1 , 300 800 1 , 400 600 700 400 2 , 100 ]

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For the following exercises, use the matrices below to perform matrix multiplication.

A = [ −1 5 3 2 ] , B = [ 3 6 4 −8 0 12 ] , C = [ 4 10 −2 6 5 9 ] , D = [ 2 −3 12 9 3 1 0 8 −10 ]

B D

[ 60 41 2 −16 120 −216 ]

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C B

[ −68 24 136 −54 −12 64 −57 30 128 ]

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For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.

A = [ 2 −5 6 7 ] , B = [ −9 6 −4 2 ] , C = [ 0 9 7 1 ] , D = [ −8 7 −5 4 3 2 0 9 2 ] , E = [ 4 5 3 7 −6 −5 1 0 9 ]

4 A + 5 D

Undefined; dimensions do not match.

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3 D + 4 E

[ −8 41 −3 40 −15 −14 4 27 42 ]

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100 D −10 E

[ −840 650 −530 330 360 250 −10 900 110 ]

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For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: A 2 = A A )

A = [ −10 20 5 25 ] , B = [ 40 10 −20 30 ] , C = [ −1 0 0 −1 1 0 ]

B A

[ −350 1 , 050 350 350 ]

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B C

Undefined; inner dimensions do not match.

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B 2

[ 1 , 400 700 −1 , 400 700 ]

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B 2 A 2

[ 332 , 500 927 , 500 −227 , 500 87 , 500 ]

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( A B ) 2

[ 490 , 000 0 0 490 , 000 ]

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For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: A 2 = A A )

A = [ 1 0 2 3 ] , B = [ −2 3 4 −1 1 −5 ] , C = [ 0.5 0.1 1 0.2 −0.5 0.3 ] , D = [ 1 0 −1 −6 7 5 4 2 1 ]

A B

[ −2 3 4 −7 9 −7 ]

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B D

[ −4 29 21 −27 −3 1 ]

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D 2

[ −3 −2 −2 −28 59 46 −4 16 7 ]

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D 3

[ 1 −18 −9 −198 505 369 −72 126 91 ]

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A ( B C )

[ 0 1.6 9 −1 ]

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Technology

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.

A = [ −2 0 9 1 8 −3 0.5 4 5 ] , B = [ 0.5 3 0 −4 1 6 8 7 2 ] , C = [ 1 0 1 0 1 0 1 0 1 ]

B A

[ 2 24 −4.5 12 32 −9 −8 64 61 ]

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B C

[ 0.5 3 0.5 2 1 2 10 7 10 ]

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Extensions

For the following exercises, use the matrix below to perform the indicated operation on the given matrix.

B = [ 1 0 0 0 0 1 0 1 0 ]

B 2

[ 1 0 0 0 1 0 0 0 1 ]

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B 4

[ 1 0 0 0 1 0 0 0 1 ]

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Using the above questions, find a formula for B n . Test the formula for B 201 and B 202 , using a calculator.

B n = { [ 1 0 0 0 1 0 0 0 1 ] , n even, [ 1 0 0 0 0 1 0 1 0 ] , n odd .

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Questions & Answers

the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
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Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
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meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
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meena
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meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
Practice Key Terms 5

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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