<< Chapter < Page Chapter >> Page >

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 ]
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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 ]

Got questions? Get instant answers now!

D B

Undidentified; dimensions do not match

Got questions? Get instant answers now!

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 ]

Got questions? Get instant answers now!

−4 C

[ −64 −12 −28 −72 −360 −20 −12 −116 ]

Got questions? Get instant answers now!

100 D

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

Got questions? Get instant answers now!

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 ]

Got questions? Get instant answers now!

C B

[ −68 24 136 −54 −12 64 −57 30 128 ]

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

3 D + 4 E

[ −8 41 −3 40 −15 −14 4 27 42 ]

Got questions? Get instant answers now!

100 D −10 E

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

Got questions? Get instant answers now!

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 ]

Got questions? Get instant answers now!

B C

Undefined; inner dimensions do not match.

Got questions? Get instant answers now!

B 2

[ 1 , 400 700 −1 , 400 700 ]

Got questions? Get instant answers now!

B 2 A 2

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

Got questions? Get instant answers now!

( A B ) 2

[ 490 , 000 0 0 490 , 000 ]

Got questions? Get instant answers now!

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 ]

Got questions? Get instant answers now!

B D

[ −4 29 21 −27 −3 1 ]

Got questions? Get instant answers now!

D 2

[ −3 −2 −2 −28 59 46 −4 16 7 ]

Got questions? Get instant answers now!

D 3

[ 1 −18 −9 −198 505 369 −72 126 91 ]

Got questions? Get instant answers now!

A ( B C )

[ 0 1.6 9 −1 ]

Got questions? Get instant answers now!

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 ]

Got questions? Get instant answers now!

B C

[ 0.5 3 0.5 2 1 2 10 7 10 ]

Got questions? Get instant answers now!

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 ]

Got questions? Get instant answers now!

B 4

[ 1 0 0 0 1 0 0 0 1 ]

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 5

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask