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 − 1 1 1 − 2 1 1 1 1
  • B 5 9 − 7 1 − 4 9 6 1
  • C − 1 − 2 1 1 1 1 1 1
  • D − 6 − 4 − 1 5 1 7

Q2:

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

  • A 0 0 3 5 − 3 5
  • BIt is not possible.
  • C 0 − 3 5
  • D [ 0 − 3 5 ]
  • E 0 3 5 0 − 3 5

Q3:

Given that 𝐴=−5−650, find 𝐴+5𝐴+30𝐼.

  • A 6 6 − 5 5 0 5 5
  • B 0 0 0 0
  • C 0 3 0 3 0 0
  • D 1 0 0 1

Q4:

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

  • A 7 5
  • B [ 7 5 ]
  • C − 1 8 1 8 2 8 − 1 6 − 3 3
  • Dit is not possible
  • E − 1 8 2 8 − 3 1 8 − 1 6 3

Q5:

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

  • A − 8 0 1 6 − 8 4 − 1 1 5 6 8 − 7 4 8 1 2 1 0 5
  • B − 8 8 3 6 4 2 8 3 2 6 3
  • C − 8 8 8 3 6 3 2 4 2 6 3
  • D − 8 0 − 1 1 5 4 8 1 6 6 8 1 2 − 8 4 − 7 1 0 5
  • EIt is not possible.

Q6:

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

Find 𝐴𝐵 if possible.

  • A − 6 2 9 − 3 0 4 4 3 2 − 1 6 − 1 9 − 2 0 1 2
  • B − 6 4 4 − 1 9 2 9 3 2 − 2 0 − 3 0 − 1 6 1 2
  • C 2 6 4 − 9 − 1 0 − 1 6 2 8 − 4 5 − 8 1 4
  • D 2 6 − 1 0 − 4 5 4 − 1 6 − 8 − 9 2 8 1 4

Q7:

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

  • A 𝐴 𝐶 𝐵 = − 3 0 − 1 8 3 0 1 8 , 𝐵 𝐴 𝐶 = − 3 0 − 1 8 3 0 1 8
  • BIt is not possible.
  • C 𝐴 𝐶 𝐵 = 0 − 3 8 0 2 3 , 𝐵 𝐴 𝐶 = − 3 0 3 0 − 1 8 1 8
  • D 𝐴 𝐶 𝐵 = 0 0 − 3 8 2 3 , 𝐵 𝐴 𝐶 = 5 3 3 0 1 8
  • E 𝐴 𝐶 𝐵 = − 3 0 3 0 − 1 8 1 8 , 𝐵 𝐴 𝐶 = 5 3 0 3 1 8

Q8:

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

Find 𝐴𝐵.

  • A 𝐴 𝐵 = 𝑎 𝑎 𝑎 𝑏 𝑏 𝑏 𝑐 𝑐 𝑐
  • B 𝐴 𝐵 = 𝑎 𝑏 𝑏 𝑐 𝑎 𝑏 𝑐 𝑐 𝑎
  • C 𝐴 𝐵 = 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐
  • D 𝐴 𝐵 = 𝑎 𝑎 𝑎 𝑏 𝑏 𝑏 𝑐 𝑐 𝑐
  • E 𝐴 𝐵 = 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐

Find 𝐵′𝐴′.

  • A 𝐵 ′ 𝐴 ′ = 𝑎 𝑎 𝑎 𝑏 𝑏 𝑏 𝑐 𝑐 𝑐
  • B 𝐵 ′ 𝐴 ′ = 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐
  • C 𝐵 ′ 𝐴 ′ = 𝑎 𝑐 𝑐 𝑏 𝑎 𝑐 𝑏 𝑏 𝑎
  • D 𝐵 ′ 𝐴 ′ = 𝑎 𝑎 𝑎 𝑏 𝑏 𝑏 𝑐 𝑐 𝑐
  • E 𝐵 ′ 𝐴 ′ = 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐

Q9:

Suppose 𝐴=[1−2−3]𝐵=81−3.and

Find the product 𝐴𝐵.

  • A [ 1 5 ]
  • B [ 6 ]
  • C [ 1 5 ]
  • D [ 1 9 ]
  • E [ 1 9 ]

Find the product 𝐵𝐴.

  • A 8 1 6 2 4 1 2 3 3 6 9
  • B 8 − 1 6 − 2 4 1 − 2 − 3 − 3 6 9
  • C 8 − 1 6 − 2 4 1 − 2 − 3 3 − 6 − 9
  • D 8 − 1 6 − 2 4 1 − 2 − 3 − 3 6 9
  • E 8 − 1 6 − 2 4 1 − 2 − 3 3 − 6 − 9

Q10:

Evaluate the matrix product 811−3710−131.

  • A 1 1 3 3 − 9 1 0
  • B 8 0 − 1 1 − 9 7
  • C 8 3 8 9 2 1 − 4
  • D 4 7 − 1 9 − 5 1 − 4
  • E 1 1 0 − 8 2 1 − 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.

  • A 𝐴 𝐵 = 1 4 − 1 6 4 − 2 , 𝐴 𝐵 = 1 8 3 6 − 4 − 6
  • B 𝐴 𝐵 = 2 6 − 4 − 4 2 , 𝐴 𝐵 = 2 6 − 4 − 4 2
  • C 𝐴 𝐵 = 1 4 4 − 1 6 − 2 , 𝐴 𝐵 = 1 4 4 − 1 6 − 2
  • D 𝐴 𝐵 = 1 4 4 − 1 6 − 2 , 𝐴 𝐵 = 1 8 − 4 3 6 − 6

Q13:

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

  • A 0 − 1 − 2 0 − 1 − 2 0 1 2 , − 1
  • B 0 1 2 0 1 2 0 − 1 − 2 , − 1
  • C 0 0 0 − 1 − 1 1 2 2 2 , 1
  • D 0 1 2 0 1 2 0 − 1 − 2 , 1
  • E 0 − 1 − 2 0 − 1 − 2 0 1 2 , 1

Q14:

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

  • A 2 7 − 3 3 − 3 6 4 2
  • B 2 7 − 3 6 − 3 3 4 2
  • C 3 − 3 6 − 9 9 0
  • D 3 − 9 − 3 6 9 0

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 − 1 0 0
  • B 1 1 0 0
  • C 0 0 1 1
  • D 0 0 − 1 − 1

Q17:

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

  • A − 2 0 0 0
  • B − 1 1 0 0
  • C 1 − 1 0 0
  • D 2 0 0 0

Q18:

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

  • A [ 3 0 ]
  • B [ − 4 1 2 1 4 ]
  • C [ 2 2 ]
  • D − 4 1 2 1 4

Q19:

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

  • A 1 0 − 1 4 − 2 3 2 2
  • Bundefined
  • C 1 0 − 2 3 − 1 4 2 2
  • D 5 − 2 2 − 1 5 1 6 − 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.

  • A 𝐽 = 1 0 0 0 1 0 , 𝐾 = 1 0 0 1 0 0 , 𝐽 and 𝐾 have different dimensions.
  • B 𝐽 = 1 1 1 1 1 1 1 1 1 , 𝐾 = 1 1 1 1 , 𝐽 and 𝐾 have different dimensions.
  • C 𝐽 = 1 0 0 1 , 𝐾 = 1 0 0 0 1 0 0 0 1 , 𝐽 and 𝐾 have different dimensions.
  • D 𝐽 = 1 1 1 1 , 𝐾 = 1 1 1 1 1 1 1 1 1 , 𝐽 and 𝐾 have different dimensions.
  • E 𝐽 = 1 0 0 0 1 0 0 0 1 , 𝐾 = 1 0 0 1 , 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.

  • A 2 × 1
  • B 1 × 2
  • C 2 × 3
  • Dundefined
  • E 3 × 1

Q24:

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

  • A 𝐵 𝐶
  • B 𝐶
  • C 𝐴
  • D 𝐴 𝐵
  • E 𝐵 𝐴

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)?

  • A 1 × 3 , 2 × 4 , 1 × 4 , 1 × 4
  • B 3 × 1 , 4 × 2 , 4 × 1 , 4 × 1
  • C 2 × 3 , 3 × 4 , 1 × 3 , 1 × 3
  • D 3 × 1 , 4 × 2 , 1 × 4 , 1 × 4
  • E 1 × 3 , 2 × 4 , 4 × 1 , 4 × 4

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