Worksheet: Finding an Unknown Matrix

In this worksheet, we will practice using operations on matrices and matrix inversion to find an unknown matrix as an introduction to solve a system of linear equations.

Q1:

Consider the matrices Find the matrix 𝑋 that satisfies the equation 𝑋 = 𝐴 + 𝐵 𝑇 𝑇 𝑇 .

  • A 9 1 3 7 1 0 9
  • B 5 7 1 5 4 3 1
  • C 9 7 1 1 0 3 9
  • D 5 1 5 3 7 4 1

Q2:

Given that what are 𝑋 and 𝑌 ?

  • A 𝑋 = 6 2 4 6 2 2 , 𝑌 = 1 2 1 4
  • B 𝑋 = 6 2 4 6 2 2 , 𝑌 = 0 8 6 3
  • C 𝑋 = 6 8 6 1 6 , 𝑌 = 0 8 6 3
  • D 𝑋 = 3 1 2 3 1 1 , 𝑌 = 3 4 3 8

Q3:

Adam guesses that any matrix , where , must be a combination of and . In other words, it must be for some numbers and . Ramy wants to challenge this, since he see that produces the same product when multiplied on either side. Help Adam by finding and so that

  • A
  • B
  • C
  • D
  • E

Q4:

Given that 𝑀 = 5 6 5 4 , find the values of 𝑥 and 𝑦 that satisfy 𝑀 + 𝑥 𝑀 + 𝑦 𝐼 = 𝑂 , where 𝑂 is the zero matrix of order 2 × 2 and 𝐼 is the unit matrix of order 2 × 2 .

  • A 𝑥 = 1 1 , 𝑦 = 1 0
  • B 𝑥 = 1 , 𝑦 = 0
  • C 𝑥 = 1 1 , 𝑦 = 0
  • D 𝑥 = 1 , 𝑦 = 1 0

Q5:

Given that find the values of 𝑥 and 𝑦 .

  • A 𝑥 = 7 , 𝑦 = 2 6
  • B 𝑥 = 8 , 𝑦 = 1 0
  • C 𝑥 = 5 , 𝑦 = 3 0
  • D 𝑥 = 1 , 𝑦 = 9

Q6:

Given that find the matrix 𝐴 .

  • A 0 3 5 4
  • B 3 3 1 4
  • C 3 2 4 7
  • D 6 3 3 4

Q7:

Given that find the matrix 𝐴 .

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

Q8:

Determine 𝑋 , given that

  • A 1 0 0 0 1 0 0 0 1
  • B 1 1 1 1
  • C 1 1 1 1 1 1 1 1 1
  • D 1 0 0 1

Q9:

Solve the matrix equation 3 𝑋 + 3 6 5 7 = 𝑋 + 5 4 1 7 .

  • A 4 1 2 0
  • B 4 2 2 1 6 2 8
  • C 7 7 7 7
  • D 2 1 1 8 1 4
  • E 2 2 2 7 1 4

Q10:

Find the matrix 𝐴 that satisfies the equation 𝐴 2 𝐴 = 5 9 9 1 𝑇 .

  • A 5 3 2 3 2 1
  • B 5 3 3 1
  • C 5 3 2 3 2 1
  • D 5 3 3 1

Q11:

Given that

solve the following matrix equation for :

  • A
  • B
  • C
  • D
  • E

Q12:

Given that find the matrix 𝑋 .

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

Q13:

Given that solve the equation

  • A 1 3 1 5 7 2 6 1 9
  • B 7 5 5 2 1 2
  • C 1 7 3 1 2 6 5 1 9
  • D 7 2 5 1 5 2

Q14:

Consider the matrices Determine the matrix 𝑋 that satisfies 𝑋 = 𝐴 + ( 𝐵 𝐶 ) 𝑇 2 𝑇 .

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

Q15:

Consider the matrices 𝐴 and 𝐵 : Find 𝐴 + 𝐵 1 2 .

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

Q16:

Given that determine the matrix 𝑋 that satisfies the relation 𝑋 = ( 𝐴 𝐵 + 𝐴 𝐶 ) 𝑇 .

  • A 1 2 4 3 4 3
  • B 1 6 0 4 0 1 3
  • C 1 6 4 0 0 1 3
  • D 1 2 4 4 3 3

Q17:

Let and . Solve the matrix equation for the matrix .

  • A
  • B
  • C
  • D
  • E

Q18:

Consider the matrices Find the matrix 𝑋 that satisfies the equation 𝑋 = 𝐴 𝐵 𝑇 𝑇 𝑇 .

  • A 4 1 0 2 0 7 9 8
  • B 1 2 7 0 3 0 0
  • C 4 7 1 0 9 2 0 8
  • D 1 2 0 0 7 3 0

Q19:

Consider the matrices Find the matrix 𝑋 that satisfies the equation 𝑋 = 𝐴 + 𝐵 𝑇 𝑇 𝑇 .

  • A 1 3 7 1 3 2 1 5 9
  • B 5 6 3 1 3 9
  • C 1 3 2 7 1 5 1 3 9
  • D 5 3 3 6 1 9

Q20:

Given that what are 𝑋 and 𝑌 ?

  • A 𝑋 = 6 2 2 1 8 1 6 , 𝑌 = 0 2 3 6
  • B 𝑋 = 3 3 4 3 2 0 , 𝑌 = 1 1 9 5 1 6
  • C 𝑋 = 1 4 7 1 2 , 𝑌 = 1 1 9 5 1 6
  • D 𝑋 = 3 1 1 9 8 , 𝑌 = 1 4 7 1 2

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