Worksheet: Matrix Multiplication

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In this worksheet, we will practice identifying the conditions for matrix multiplication and evaluating the product of two matrices if possible.

Q1:

Given that 𝐴=−5−5−66𝐵=46−35,, find (𝐴+𝐵)𝐴.

  • A−111−21111
  • B59−71−4961
  • C−1−2111111
  • D−6−4−1517

Q2:

Given that 𝐴=−77𝐵=[0−5],, find 𝐴𝐵 if possible.

  • A0035−35
  • BIt is not possible.
  • C0−35
  • D[0−35]
  • E0350−35

Q3:

Given that 𝐴=−5−650, find 𝐴+5𝐴+30𝐼.

  • A66−55055
  • B0000
  • C030300
  • D1001

Q4:

Given that 𝐴=−3−7−1341𝐵=6−43,, find 𝐴𝐵 if possible.

  • A75
  • B[75]
  • C−181828−16−33
  • Dit is not possible
  • E−1828−318−163

Q5:

Consider the matrices 𝐴=11−2−4477,𝐵=−8−96−489. Find 𝐴𝐵, if possible.

  • A−8016−84−11568−74812105
  • B−88364283263
  • C−88836324263
  • D−80−11548166812−84−7105
  • EIt is not possible.

Q6:

Consider the matrices 𝐴=−3−44−4445−1−1,𝐵=−2−3260235−4.

Find 𝐴𝐵 if possible.

  • A−629−304432−16−19−2012
  • B−644−192932−20−30−1612
  • C264−9−10−1628−45−814
  • D26−10−454−16−8−92814

Q7:

Consider the matrices 𝐴=01,𝐵=−41−66,𝐶=[53]. Find 𝐴𝐶𝐵 and 𝐵𝐴𝐶 if possible.

  • A𝐴𝐶𝐵=−30−183018, 𝐵𝐴𝐶=−30−183018
  • BIt is not possible.
  • C𝐴𝐶𝐵=0−38023, 𝐵𝐴𝐶=−3030−1818
  • D𝐴𝐶𝐵=00−3823, 𝐵𝐴𝐶=533018
  • E𝐴𝐶𝐵=−3030−1818, 𝐵𝐴𝐶=530318

Q8:

Consider the matrices 𝐴=111,𝐴′=[111],𝐵=[𝑎𝑏𝑐],𝐵′=𝑎𝑏𝑐.

Find 𝐴𝐵.

  • A𝐴𝐵=𝑎𝑎𝑎𝑏𝑏𝑏𝑐𝑐𝑐
  • B𝐴𝐵=𝑎𝑏𝑏𝑐𝑎𝑏𝑐𝑐𝑎
  • C𝐴𝐵=𝑎𝑏𝑐𝑎𝑏𝑐𝑎𝑏𝑐
  • D𝐴𝐵=𝑎𝑎𝑎𝑏𝑏𝑏𝑐𝑐𝑐
  • E𝐴𝐵=𝑎𝑏𝑐𝑎𝑏𝑐𝑎𝑏𝑐

Find 𝐵′𝐴′.

  • A𝐵′𝐴′=𝑎𝑎𝑎𝑏𝑏𝑏𝑐𝑐𝑐
  • B𝐵′𝐴′=𝑎𝑏𝑐𝑎𝑏𝑐𝑎𝑏𝑐
  • C𝐵′𝐴′=𝑎𝑐𝑐𝑏𝑎𝑐𝑏𝑏𝑎
  • D𝐵′𝐴′=𝑎𝑎𝑎𝑏𝑏𝑏𝑐𝑐𝑐
  • E𝐵′𝐴′=𝑎𝑏𝑐𝑎𝑏𝑐𝑎𝑏𝑐

Q9:

Suppose 𝐴=[1−2−3]𝐵=81−3.and

Find the product 𝐴𝐵.

  • A[15]
  • B[6]
  • C[15]
  • D[19]
  • E[19]

Find the product 𝐵𝐴.

  • A81624123369
  • B8−16−241−2−3−369
  • C8−16−241−2−33−6−9
  • D8−16−241−2−3−369
  • E8−16−241−2−33−6−9

Q10:

Evaluate the matrix product 811−3710−131.

  • A1133−910
  • B80−11−97
  • C838921−4
  • D47−19−51−4
  • E110−821−3

Q11:

Consider the matrix product 5−2−2273814616.

What can you conclude about it?

  • AFor a given 2×3 matrix 𝐴, there cannot be any matrix 𝐵 except the 2×2 identity matrix for which 𝐵𝐴=𝐴.
  • BFor a given 2×3 matrix 𝐴, there can be a matrix 𝐵 that is not the 3×3 identity matrix for which 𝐴𝐵=𝐴.
  • CFor a given 2×3 matrix 𝐴, there can be a matrix 𝐵 that is not the 2×2 identity matrix for which 𝐵𝐴=𝐴.
  • DFor a given 2×3 matrix 𝐴, there can be a matrix 𝐵 that is not the 3×3 identity matrix for which 𝐴𝐵=𝐵.

Is it possible to find a matrix 𝐵 with the above property for every 2×3 matrix 𝐴?

  • Ano
  • Byes

Q12:

Consider the matrices 𝐴=−42−6−6,𝐵=−5−110.

Find 𝐴𝐵 and 𝐴𝐵.

  • A𝐴𝐵=14−164−2, 𝐴𝐵=1836−4−6
  • B𝐴𝐵=26−4−42, 𝐴𝐵=26−4−42
  • C𝐴𝐵=144−16−2, 𝐴𝐵=144−16−2
  • D𝐴𝐵=144−16−2, 𝐴𝐵=18−436−6

Q13:

Let 𝑥=[−1−11] and 𝑦=[012]. Find 𝑥𝑦 and 𝑥𝑦.

  • A0−1−20−1−2012, −1
  • B0120120−1−2, −1
  • C000−1−11222, 1
  • D0120120−1−2, 1
  • E0−1−20−1−2012, 1

Q14:

Consider the matrices 𝐴=120−3,𝐵=4−5−56,𝐶=3630. Find 𝐴𝐵𝐶 if possible.

  • A27−33−3642
  • B27−36−3342
  • C3−36−990
  • D3−9−3690

Q15:

Given that 𝐴=−1505𝐵=5−50−1,, and 𝐼 is the unit matrix of the same order, find 𝑋 for which 𝐴𝐵=𝑋×𝐼.

Q16:

Given that 𝐴=𝜃𝜃𝜃−𝜃𝐵=𝜃𝜃−𝜃−𝜃,cossinsincos,sinsincoscos find 𝐴𝐵 if possible.

  • A−1−100
  • B1100
  • C0011
  • D00−1−1

Q17:

Given that 𝐴=𝑖𝑖00𝐵=𝑖𝑖00,, and 𝑖=−1, find 𝐴𝐵 if possible.

  • A−2000
  • B−1100
  • C1−100
  • D2000

Q18:

Consider the matrices 𝐴=[12−7],𝐵=−46−2. Find 𝐴𝐵, if possible.

  • A[30]
  • B[−41214]
  • C[22]
  • D−41214

Q19:

Given that 𝐴=512−3−4−3𝐵=1−25−4,, determine 𝐴𝐵 if possible.

  • A10−14−2322
  • Bundefined
  • C10−23−1422
  • D5−22−1516−3

Q20:

Is it possible to have a 2×1 matrix and a 1×2 matrix such that 𝐴𝐵=1001? If so, give an example.

  • Ayes, 𝐴=01; 𝐵=[10]
  • Bno
  • Cyes, 𝐴=10; 𝐵=[10]

Q21:

Suppose the matrix product 𝐴𝐵𝐶 makes sense. We also know that 𝐴 has 2 rows, 𝐶 has 3 columns, and 𝐵 has 4 entries. Is it possible to determine the possible sizes of these matrices? If so, what are the possible sizes of 𝐴, 𝐵, and 𝐶?

  • Ayes, 1×2, 2×2, 3×1; 2×1, 1×5, 5×3; 2×4, 4×1, 1×3
  • Byes, 2×1, 1×4, 4×3; 2×2, 2×2, 2×3; 2×4, 4×1, 1×3
  • Cyes, 2×1, 1×4, 4×3; 2×2, 2×4, 4×3; 2×4, 4×1, 1×3
  • Dno
  • Eyes, 1×2, 2×2, 3×1; 2×2, 2×2, 2×3; 4×2, 2×3, 3×1

Q22:

Find the matrices 𝐽 and 𝐾 such that, for any 2×3 matrix 𝑋, 𝐽𝑋=𝑋 and 𝑋𝐾=𝑋. Explain why 𝐽 and 𝐾 are not the same.

  • A𝐽=100010, 𝐾=100100, 𝐽 and 𝐾 have different dimensions.
  • B𝐽=111111111, 𝐾=1111, 𝐽 and 𝐾 have different dimensions.
  • C𝐽=1001, 𝐾=100010001, 𝐽 and 𝐾 have different dimensions.
  • D𝐽=1111, 𝐾=111111111, 𝐽 and 𝐾 have different dimensions.
  • E𝐽=100010001, 𝐾=1001, 𝐽 and 𝐾 have different dimensions.

Q23:

Given that 𝐴 is a matrix of order 2×3 and 𝐵 is a matrix of order 1×3, find the order of the matrix 𝐴𝐵, if possible.

  • A2×1
  • B1×2
  • C2×3
  • Dundefined
  • E3×1

Q24:

Suppose 𝐴=11−10−21,𝐵=−21−30,𝐶=0−23110.and Which of the following products is defined?

  • A𝐵𝐶
  • B𝐶
  • C𝐴
  • D𝐴𝐵
  • E𝐵𝐴

Q25:

Suppose 𝐴 is a 1×2 matrix, 𝐵 is a 2×3 matrix, and 𝐶 is a 3×4 matrix. What are the sizes of the product matrices 𝐴𝐵,𝐵𝐶,(𝐴𝐵)𝐶, and 𝐴(𝐵𝐶)?

  • A1×3,2×4,1×4,1×4
  • B3×1,4×2,4×1,4×1
  • C2×3,3×4,1×3,1×3
  • D3×1,4×2,1×4,1×4
  • E1×3,2×4,4×1,4×4

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