Worksheet: Multiplying Matrices

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In this worksheet, we will practice multiplying matrices.

Q1:

Given that find ( 𝐴 + 𝐵 ) 𝐴 .

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

Q2:

Given that find 𝐴 2 .

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

Q3:

Given that find 𝐴 𝐵 if possible.

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

Q4:

Given that 𝐴 = − 5 − 6 5 0 , find 𝐴 + 5 𝐴 + 3 0 𝐼 2 .

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

Q5:

Determine the values of 𝑥 , 𝑦 , and 𝑧 that satisfy the following:

  • A 𝑥 = − 1 3 , 𝑦 = 8 , 𝑧 = − 1 4
  • B 𝑥 = − 3 , 𝑦 = 2 0 , 𝑧 = − 4
  • C 𝑥 = 1 , 𝑦 = 2 0 , 𝑧 = − 1 4
  • D 𝑥 = 1 , 𝑦 = 4 , 𝑧 = − 9

Q6:

Given that find 𝐴 𝐵 if possible.

  • A − 1 8 1 8 2 8 − 1 6 − 3 3
  • B ( 7 5 )
  • C − 1 8 2 8 − 3 1 8 − 1 6 3
  • D 7 5
  • E it is not possible

Q7:

Consider the matrices 𝐴 = 1 1 − 2 − 4 4 7 7 , 𝐵 = − 8 − 9 6 − 4 8 9 . Find 𝐴 𝐵 , if possible.

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

Q8:

Consider the matrices

Find if possible.

  • A
  • B
  • C
  • D

Q9:

Consider the matrices Find 𝐴 𝐶 𝐵 and 𝐵 𝐴 𝐶 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
  • B 𝐴 𝐶 𝐵 = − 3 0 3 0 − 1 8 1 8 , 𝐵 𝐴 𝐶 = 5 3 0 3 1 8
  • 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
  • EIt is not possible.

Q10:

Consider the matrices

Find .

  • A
  • B
  • C
  • D
  • E

Find .

  • A
  • B
  • C
  • D
  • E

Q11:

Suppose

Find the product .

  • A
  • B
  • C
  • D
  • E

Find the product .

  • A
  • B
  • C
  • D
  • E

Q12:

Evaluate the matrix product

  • A
  • B
  • C
  • D
  • E

Q13:

Consider the matrix product

What can you conclude about it?

  • A For a given 2 × 3 matrix 𝐴 , there can be a matrix 𝐵 that is not the 3 × 3 identity matrix for which 𝐴 𝐵 = 𝐵 .
  • BFor a given 2 × 3 matrix 𝐴 , there can be a matrix 𝐵 that is not the 3 × 3 identity matrix for which 𝐴 𝐵 = 𝐴 .
  • CFor a given 2 × 3 matrix 𝐴 , there cannot be any matrix 𝐵 except the 2 × 2 identity matrix for which 𝐵 𝐴 = 𝐴 .
  • DFor a given 2 × 3 matrix 𝐴 , there can be a matrix 𝐵 that is not the 2 × 2 identity matrix for which 𝐵 𝐴 = 𝐴 .

Is it possible to find a matrix 𝐵 with the above property for every 2 × 3 matrix 𝐴 ?

  • Ano
  • Byes

Q14:

Consider the matrices

Find 𝐴 𝐵 𝑇 and 𝐴 𝐵 𝑇 .

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

Q15:

Let 𝑥 = ( − 1 − 1 1 ) and 𝑦 = ( 0 1 2 ) . Find 𝑥 𝑦 𝑇 and 𝑥 𝑦 𝑇 .

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

Q16:

Consider the matrices Find 𝐴 𝐵 𝐶 if possible.

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

Q17:

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

Q18:

Given that find 𝐴 𝐵 if possible.

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

Q19:

Given that and 𝑖 = − 1 2 , find 𝐴 𝐵 if possible.

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

Q20:

Consider the matrices Find 𝐴 𝐵 , if possible.

  • A ( − 4 1 2 1 4 )
  • B ( 3 0 )
  • C − 4 1 2 1 4
  • D ( 2 2 )

Q21:

Given that determine 𝐴 𝐵 if possible.

  • A − 1 5 − 6 − 1 4 5
  • B − 1 5 4 2 2 5
  • C − 1 5 4 2 1 0 5
  • Dundefined

Q22:

Given that find the value of √ 𝑥 𝑦 .

  • A 6 √ 2
  • B 9 √ 5
  • C 6 √ 1 0
  • D9

Q23:

Let , .

Verify that if matrix satisfies , then and . Find matrices and so that , and then use this to find and so that

  • A , .
  • B , .
  • C , .
  • D , .
  • E , .

Q24:

Find the values of 𝑥 and 𝑦 given

  • A 𝑥 = 7 , 𝑦 = − 1 2
  • B 𝑥 = 1 8 , 𝑦 = − 4 0
  • C 𝑥 = 5 , 𝑦 = − 4 0
  • D 𝑥 = 5 , 𝑦 = 0

Q25:

Given that where 𝑂 is the zero matrix of order 2 × 2 , find the values of 𝑥 and 𝑦 .

  • A 𝑥 = − 3 , 𝑦 = − 1
  • B 𝑥 = − 4 , 𝑦 = 1
  • C 𝑥 = − 8 , 𝑦 = − 9
  • D 𝑥 = 1 , 𝑦 = − 9

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