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Worksheet: Matrix Operations

In this worksheet, we will practice performing operations on matrices, such as adding them, subtracting them, transposing them, and multiplying them by scalars.

Q1:

Suppose that 𝐴 =  1 βˆ’ 3 βˆ’ 4 2  , 𝐡 =  2 0 1 βˆ’ 1  , and 𝐢 =  0 1 βˆ’ 3 0  .

Find 𝐴 𝐡 .

  • A  3 βˆ’ 3 βˆ’ 3 1 
  • B  2 βˆ’ 6 5 βˆ’ 5 
  • C  βˆ’ 4 2 βˆ’ 3 9 
  • D  βˆ’ 1 3 βˆ’ 6 βˆ’ 2 
  • E  9 1 βˆ’ 6 βˆ’ 4 

Find 𝐴 𝐢 .

  • A  9 1 βˆ’ 6 βˆ’ 4 
  • B  1 βˆ’ 2 βˆ’ 7 2 
  • C  3 βˆ’ 3 βˆ’ 3 1 
  • D  2 βˆ’ 6 5 βˆ’ 5 
  • E  βˆ’ 1 3 βˆ’ 6 βˆ’ 2 

Find 𝐴 ( 2 𝐡 + 7 𝐢 ) .

  • A  βˆ’ 2 1 3 βˆ’ 3 3 βˆ’ 4 
  • B  8 4 βˆ’ 1 2 βˆ’ 6 
  • C  6 1 1 3 βˆ’ 5 4 βˆ’ 3 2 
  • D  βˆ’ 2 4 2 βˆ’ 1 1 5 3 
  • E  βˆ’ 2 βˆ’ 4 2 4 

Express 𝐴 ( 2 𝐡 + 7 𝐢 ) in terms of 𝐴 𝐡 and 𝐴 𝐢 .

  • A 2 𝐴 𝐡 + 7 𝐴 𝐢
  • B 2 𝐡 + 7 𝐴 𝐢
  • C 2 𝐡 𝐴 + 7 𝐢
  • D 2 𝐴 𝐡 + 7 𝐢
  • E 2 𝐡 𝐴 + 7 𝐢 𝐴

Q2:

If and then which of the following cannot exist?

  • A 𝐴 βˆ’ 𝐡
  • B 3 𝐴 βˆ’ 𝐡
  • C ο€Ή 𝐡 𝐴   
  • D 𝐴 𝐡

Q3:

Given that find 𝐴 𝑇 .

  • A  βˆ’ 6 4 6 8 βˆ’ 2 1 
  • B  1 βˆ’ 2 8 6 4 βˆ’ 6 
  • C  1 8 4 βˆ’ 2 6 βˆ’ 6 
  • D  βˆ’ 2 1 6 8 βˆ’ 6 4 

Q4:

Given that determine .

  • A
  • B
  • C
  • D

Q5:

Given that determine the matrix .

  • A
  • B
  • C
  • D

Q6:

Given that the order of the matrix 𝐴 is 2 Γ— 2 , and that of the matrix 𝐡 𝑇 is 1 Γ— 2 , which of the following operations can be performed?

  • A 𝐴 + 𝐡 𝑇 𝑇
  • B 𝐴 + 𝐡
  • C 𝐴 Γ— 𝐡 𝑇
  • D 𝐴 Γ— 𝐡

Q7:

𝐽 and 𝐾 are two matrices with the property that for any 3 Γ— 3 matrix 𝑋 , 𝐽 𝑋 = 𝑋 and 𝑋 𝐾 = 𝑋 . Are 𝐽 and 𝐾 equal?

  • A No, they are different matrices of the same dimensions.
  • B No, they have different dimensions.
  • C Yes, they are both the 3 Γ— 3 identity matrix.

Q8:

Consider the matrices and . Find .

  • A
  • B
  • C
  • D
  • E

Q9:

𝐽 and 𝐾 are two matrices with the property that for any 2 Γ— 3 matrix 𝑋 , 𝐽 𝑋 = 𝑋 and 𝑋 𝐾 = 𝑋 . Are 𝐽 and 𝐾 equal?

  • A No, they are different matrices of the same dimensions.
  • B Yes, they are both the identity matrix.
  • C No, they have different dimensions.

Q10:

Evaluate

  • A  1 8 7 βˆ’ 3 βˆ’ 2 
  • B  1 8 βˆ’ 4 1 4 8 
  • C  2 1 1 0 1 0 8 
  • D  1 8 1 0 0 8 
  • E  1 8 1 0 βˆ’ 6 8 

Q11:

Suppose that 𝐴 =  1 0 0 0  and 𝐡 =  π‘Ž 𝑏 𝑐 𝑑  . Solve the equations from 𝐴 𝐡 = 𝐡 𝐴 to find conditions on π‘Ž , 𝑏 , 𝑐 , and 𝑑 for this equality to be true.

  • A 𝑏 = π‘Ž + 𝑐 , 𝑑 = 0
  • B π‘Ž = 𝑏 + 𝑑 , 𝑐 = 2 π‘Ž
  • C 𝑏 = π‘Ž + 𝑐 , 𝑑 = 2 𝑏
  • D π‘Ž = 𝑏 + 𝑑 , 𝑐 = 0
  • E π‘Ž = 𝑏 + 𝑑 , 𝑐 = π‘Ž

Q12:

Given that find the result of , if possible.

  • A
  • BIt is not possible.
  • C
  • D
  • E

Q13:

Consider the matrix We wish to find the matrix 𝑋 for which 𝐴 𝑋 = 𝐼 , where 𝐼 is the 3 Γ— 3 identity matrix. Suppose that

Find ( 𝐴 𝑋 ) 3 , 1 in terms of π‘Ž , 𝑏 , … , 𝑖 . Hence, by comparing this to 𝐼 3 , 1 , find the value of 𝑔 .

  • A 𝑔 = βˆ’ 2
  • B 𝑔 = 1
  • C 𝑔 = 1 3
  • D 𝑔 = 0
  • E 𝑔 = βˆ’ 1 3

By considering suitable entries of 𝐴 𝑋 , find the values of β„Ž and 𝑖 .

  • A β„Ž = 0 , 𝑖 = 1 3
  • B β„Ž = βˆ’ 1 3 , 𝑖 = 0
  • C β„Ž = 0 , 𝑖 = βˆ’ 1 3
  • D β„Ž = 1 3 , 𝑖 = 0
  • E β„Ž = 1 , 𝑖 = βˆ’ 2

Using your answers to the previous parts of the question, find the values of 𝑓 and 𝑐 .

  • A 𝑓 = 1 , 𝑐 = βˆ’ 2
  • B 𝑓 = βˆ’ 1 3 , 𝑐 = 1 3
  • C 𝑓 = 1 3 , 𝑐 = 1 3
  • D 𝑓 = 1 3 , 𝑐 = βˆ’ 1 3
  • E 𝑓 = βˆ’ 1 3 , 𝑐 = βˆ’ 1 3

Find the values of 𝑑 , π‘Ž , 𝑒 , and 𝑏 .

  • A 𝑑 = 0 , π‘Ž = 1 , 𝑒 = 1 , 𝑏 = βˆ’ 2
  • B 𝑑 = 0 , π‘Ž = 0 , 𝑒 = 1 , 𝑏 = βˆ’ 2
  • C 𝑑 = 0 , π‘Ž = 1 , 𝑒 = 1 3 , 𝑏 = βˆ’ 2
  • D 𝑑 = 1 , π‘Ž = 1 , 𝑒 = 1 , 𝑏 = βˆ’ 2
  • E 𝑑 = 0 , π‘Ž = 1 3 , 𝑒 = 1 , 𝑏 = βˆ’ 2

Q14:

If 𝐴 =  1 βˆ’ 3 βˆ’ 4 2  and 𝐡 =  2 0 1 βˆ’ 1  , is ( 7 𝐴 ) 𝐡 = 𝐴 ( 7 𝐡 ) ?

  • Ayes
  • Bno

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