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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

find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
Give me the reciprocal of even number
Aliyu
The reciprocal of an even number is a proper fraction
Jamilu
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
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Practice Key Terms 5

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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