Worksheet: Matrix Multiplication

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=5566𝐵=4635,, find (𝐴+𝐵)𝐴.

  • A11121111
  • B59714961
  • C12111111
  • D641517

Q2:

Given that 𝐴=77𝐵=[05],, find 𝐴𝐵 if possible.

  • A003535
  • BIt is not possible.
  • C035
  • D[035]
  • E035035

Q3:

Given that 𝐴=5650, find 𝐴+5𝐴+30𝐼.

  • A6655055
  • B0000
  • C030300
  • D1001

Q4:

Given that 𝐴=371341𝐵=643,, find 𝐴𝐵 if possible.

  • A75
  • B[75]
  • C1818281633
  • Dit is not possible
  • E1828318163

Q5:

Consider the matrices 𝐴=1124477,𝐵=896489. Find 𝐴𝐵, if possible.

  • A8016841156874812105
  • B88364283263
  • C88836324263
  • D8011548166812847105
  • EIt is not possible.

Q6:

Consider the matrices 𝐴=344444511,𝐵=232602354.

Find 𝐴𝐵 if possible.

  • A62930443216192012
  • B64419293220301612
  • C264910162845814
  • D261045416892814

Q7:

Consider the matrices 𝐴=01,𝐵=4166,𝐶=[53]. Find 𝐴𝐶𝐵 and 𝐵𝐴𝐶 if possible.

  • A𝐴𝐶𝐵=30183018, 𝐵𝐴𝐶=30183018
  • BIt is not possible.
  • C𝐴𝐶𝐵=038023, 𝐵𝐴𝐶=30301818
  • D𝐴𝐶𝐵=003823, 𝐵𝐴𝐶=533018
  • E𝐴𝐶𝐵=30301818, 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 𝐵𝐴.

  • A𝐵𝐴=𝑎𝑎𝑎𝑏𝑏𝑏𝑐𝑐𝑐
  • B𝐵𝐴=𝑎𝑏𝑐𝑎𝑏𝑐𝑎𝑏𝑐
  • C𝐵𝐴=𝑎𝑐𝑐𝑏𝑎𝑐𝑏𝑏𝑎
  • D𝐵𝐴=𝑎𝑎𝑎𝑏𝑏𝑏𝑐𝑐𝑐
  • E𝐵𝐴=𝑎𝑏𝑐𝑎𝑏𝑐𝑎𝑏𝑐

Q9:

Suppose 𝐴=[123]𝐵=813.and

Find the product 𝐴𝐵.

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

Find the product 𝐵𝐴.

  • A81624123369
  • B81624123369
  • C81624123369
  • D81624123369
  • E81624123369

Q10:

Evaluate the matrix product 8113710131.

  • A1133910
  • B801197
  • C8389214
  • D4719514
  • E1108213

Q11:

Consider the matrix product 522273814616.

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=4266,𝐵=5110.

Find 𝐴𝐵 and 𝐴𝐵.

  • A𝐴𝐵=141642, 𝐴𝐵=183646
  • B𝐴𝐵=26442, 𝐴𝐵=26442
  • C𝐴𝐵=144162, 𝐴𝐵=144162
  • D𝐴𝐵=144162, 𝐴𝐵=184366

Q13:

Let 𝑥=[111] and 𝑦=[012]. Find 𝑥𝑦 and 𝑥𝑦.

  • A012012012, 1
  • B012012012, 1
  • C000111222, 1
  • D012012012, 1
  • E012012012, 1

Q14:

Consider the matrices 𝐴=1203,𝐵=4556,𝐶=3630. Find 𝐴𝐵𝐶 if possible.

  • A27333642
  • B27363342
  • C336990
  • D393690

Q15:

Given that 𝐴=1505𝐵=5501,, and 𝐼 is the unit matrix of the same order, find 𝑋 for which 𝐴𝐵=𝑋×𝐼.

Q16:

Given that 𝐴=𝜃𝜃𝜃𝜃𝐵=𝜃𝜃𝜃𝜃,cossinsincos,sinsincoscos find 𝐴𝐵 if possible.

  • A1100
  • B1100
  • C0011
  • D0011

Q17:

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

  • A2000
  • B1100
  • C1100
  • D2000

Q18:

Consider the matrices 𝐴=[127],𝐵=462. Find 𝐴𝐵, if possible.

  • A[30]
  • B[41214]
  • C[22]
  • D41214

Q19:

Given that 𝐴=512343𝐵=1254,, determine 𝐴𝐵 if possible.

  • A10142322
  • Bundefined
  • C10231422
  • D52215163

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=111021,𝐵=2130,𝐶=023110.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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