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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

12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
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
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
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
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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