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Worksheet: Matrix Multiplication Dimensions

Q1:

Is it possible to have a 2 Γ— 1 matrix and a 1 Γ— 2 matrix such that 𝐴 𝐡 = ο€Ό 1 0 0 1  ? If so, give an example.

  • Ayes, 𝐴 = ο€Ό 0 1  ; 𝐡 = ( 1 0 )
  • Byes, 𝐴 = ο€Ό 1 0  ; 𝐡 = ( 1 0 )
  • Cno

Q2:

Given that 𝐴 is a matrix of order 2 Γ— 3 and 𝐡 𝑇 is a matrix of order 1 Γ— 3 , find the order of the matrix 𝐴 𝐡 , if possible.

  • A 2 Γ— 3
  • B 1 Γ— 2
  • C 3 Γ— 1
  • D 2 Γ— 1
  • Eundefined

Q3:

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

  • A yes, 1 Γ— 2 , 2 Γ— 2 , 3 Γ— 1 ; 2 Γ— 2 , 2 Γ— 2 , 2 Γ— 3 ; 4 Γ— 2 , 2 Γ— 3 , 3 Γ— 1
  • Bno
  • C yes, 2 Γ— 1 , 1 Γ— 4 , 4 Γ— 3 ; 2 Γ— 2 , 2 Γ— 4 , 4 Γ— 3 ; 2 Γ— 4 , 4 Γ— 1 , 1 Γ— 3
  • D yes, 2 Γ— 1 , 1 Γ— 4 , 4 Γ— 3 ; 2 Γ— 2 , 2 Γ— 2 , 2 Γ— 3 ; 2 Γ— 4 , 4 Γ— 1 , 1 Γ— 3
  • E yes, 1 Γ— 2 , 2 Γ— 2 , 3 Γ— 1 ; 2 Γ— 1 , 1 Γ— 5 , 5 Γ— 3 ; 2 Γ— 4 , 4 Γ— 1 , 1 Γ— 3

Q4:

Find the matrices and such that, for any matrix , and . Explain why and are not the same.

  • A , , and have different dimensions.
  • B , , and have different dimensions.
  • C , , and have different dimensions.
  • D , , and have different dimensions.
  • E , , and have different dimensions.

Q5:

Suppose Which of the following products is defined?

  • A 𝐴 2
  • B 𝐴 𝐡
  • C 𝐡 𝐢
  • D 𝐡 𝐴
  • E 𝐢 2

Q6:

Suppose 𝐴 is a 1 Γ— 2 matrix, 𝐡 is a 2 Γ— 3 matrix, and 𝐢 is a 3 Γ— 4 matrix. What are the sizes of the product matrices 𝐴 𝐡 , 𝐡 𝐢 , ( 𝐴 𝐡 ) 𝐢 , and 𝐴 ( 𝐡 𝐢 ) ?

  • A 1 Γ— 3 , 2 Γ— 4 , 4 Γ— 1 , 4 Γ— 4
  • B 3 Γ— 1 , 4 Γ— 2 , 4 Γ— 1 , 4 Γ— 1
  • C 3 Γ— 1 , 4 Γ— 2 , 1 Γ— 4 , 1 Γ— 4
  • D 1 Γ— 3 , 2 Γ— 4 , 1 Γ— 4 , 1 Γ— 4
  • E 2 Γ— 3 , 3 Γ— 4 , 1 Γ— 3 , 1 Γ— 3

Q7:

If 𝐴 is a matrix of order 1 Γ— 1 and 𝐴 𝐡 is a matrix of order 1 Γ— 1 , then what is the order of 𝐡 ?

  • A 1 Γ— 2
  • B 1 Γ— 3
  • C 2 Γ— 2
  • D 1 Γ— 1

Q8:

Find a matrix such that for all matrices .

  • A
  • B
  • C
  • D
  • E

Q9:

Given that 𝐴 is a matrix of order π‘š Γ— 𝑛 , and 𝐡 is a matrix of order π‘Ÿ Γ— 𝑙 , determine the condition under which AB is defined.

  • A π‘š = 𝑛
  • B 𝑛 β‰  π‘Ÿ
  • C π‘š β‰  𝑛
  • D 𝑛 = π‘Ÿ
  • E π‘Ÿ = 𝑙

Q10:

Given that 𝐴 is a matrix of order 3 Γ— 3 and 𝐡 𝑇 is a matrix of order 2 Γ— 1 , find the order of the matrix 𝐴 𝐡 , if possible.

  • A 3 Γ— 3
  • B 2 Γ— 3
  • C 1 Γ— 2
  • Dundefined
  • E 3 Γ— 2

Q11:

Given that 𝐴 is a matrix of order 1 Γ— 2 and 𝐡 is a matrix of order 2 Γ— 3 , find the order of the matrix 𝐴 𝐡 , if possible.

  • A 1 Γ— 2
  • B 3 Γ— 1
  • C 2 Γ— 3
  • D 1 Γ— 3
  • Eundefined

Q12:

If 𝐴 is a matrix of order 1 Γ— 3 and 𝐴 𝐡 is a matrix of order 1 Γ— 3 , then what is the order of 𝐡 ?

  • A 1 Γ— 1
  • B 1 Γ— 2
  • C 3 Γ— 1
  • D 3 Γ— 3

Q13:

If 𝐴 is a matrix of order 3 Γ— 2 and 𝐴 𝐡 is a matrix of order 3 Γ— 3 , then what is the order of 𝐡 ?

  • A 3 Γ— 2
  • B 3 Γ— 1
  • C 1 Γ— 1
  • D 2 Γ— 3