Worksheet: Properties of Matrix Multiplication

In this worksheet, we will practice identifying the properties of the multiplication of matrices and comparing them to the properties of multiplication of numbers.

Q1:

Given that 𝐴=4224,𝐵=3311, find 𝐴𝐵 and 𝐵𝐴.

  • A 𝐴 𝐵 = 1 0 1 4 2 1 0 , 𝐵 𝐴 = 6 6 6 6
  • B 𝐴 𝐵 = 1 0 2 1 4 1 0 , 𝐵 𝐴 = 6 6 6 6
  • C 𝐴 𝐵 = 1 0 1 4 2 1 0 , 𝐵 𝐴 = 1 0 1 4 2 1 0
  • D 𝐴 𝐵 = 6 6 6 6 , 𝐵 𝐴 = 6 6 6 6

Q2:

Matrices 𝐴,𝐵,𝐶, and 𝐷 are square matrices. Use the associative law for three square matrices to determine which of the following proves that 𝐴(𝐵(𝐶𝐷))=((𝐴𝐵)𝐶)𝐷.

  • A 𝐴 ( 𝐵 ( 𝐶 𝐷 ) ) = ( 𝐴 ( 𝐵 𝐶 ) ) 𝐷 = 𝐴 ( ( 𝐵 𝐶 ) 𝐷 ) = ( ( 𝐴 𝐵 ) 𝐶 ) 𝐷
  • B 𝐴 ( 𝐵 ( 𝐶 𝐷 ) ) = 𝐴 ( ( 𝐵 𝐶 ) 𝐷 ) = ( 𝐴 ( 𝐵 𝐶 ) ) 𝐷 = ( ( 𝐴 𝐵 ) 𝐶 ) 𝐷
  • C 𝐴 ( 𝐵 ( 𝐶 𝐷 ) ) = 𝐴 ( ( 𝐵 𝐶 ) 𝐷 ) = ( 𝐴 ( 𝐵 + 𝐶 ) ) 𝐷 = ( ( 𝐴 + 𝐵 ) 𝐶 ) 𝐷
  • D 𝐴 ( 𝐵 ( 𝐶 𝐷 ) ) = ( 𝐴 ( 𝐵 𝐶 ) 𝐷 ) = ( ( 𝐴 𝐵 ) 𝐶 ) 𝐷

Q3:

Consider the 2×2 matrices 𝐴=1100 and 𝐵=0101. Is 𝐴𝐵=𝐵𝐴?

  • ANo
  • BYes

Q4:

Given the 1×1 matrices 𝐴=[3] and 𝐵=[4], is 𝐴𝐵=𝐵𝐴?

  • Ayes
  • Bno

Q5:

Given the 2×2 matrices 𝐴=8312 and 𝐵=8312, is 𝐴𝐵=𝐵𝐴?

  • Ano
  • Byes

Q6:

Given that 2×2 matrices 𝐴=1342 and 𝐵=1391216, is 𝐴𝐵=𝐵𝐴?

  • Ayes
  • Bno

Q7:

State whether the following statement is true or false: If 𝐴 and 𝐵 are both 2×2 matrices, then 𝐴𝐵 is never the same as 𝐵𝐴.

  • Afalse
  • Btrue

Q8:

Is there a 2×2 matrix 𝐴, other than the identity matrix 𝐼, where 𝐴𝑋=𝑋𝐴 for every 2×2 matrix 𝑋?

  • AYes
  • BNo

Q9:

Given three matrices 𝐴,𝐵, and 𝐶, which of the following is equivalent to 𝐴(𝐵+𝐶)?

  • A 𝐵 𝐴 + 𝐶 𝐴
  • B 𝐴 𝐵 + 𝐶
  • C 𝐵 + 𝐴 𝐶
  • D 𝐵 𝐴 + 𝐶
  • E 𝐴 𝐵 + 𝐴 𝐶

Q10:

State whether the following statement is true or false: If 𝐴 is a 2×3 matrix and 𝐵 and 𝐶 are 3×2 matrices, then 𝐴(𝐵+𝐶)=𝐴𝐶+𝐴𝐵.

  • Atrue
  • Bfalse

Q11:

Suppose that 𝐴=2105, 𝐵=01 and 𝐶=13.

Find 𝐴𝐵.

  • A 2 5
  • B 1 5
  • C 0 5
  • D 1 5
  • E 2 5

Find 𝐴𝐶.

  • A 2 1 5
  • B 5 1 5
  • C 2 1 5
  • D 1 1 5
  • E 2 1 6

Find 𝐴(𝐵+𝐶).

  • A 1 4
  • B 4 2 0
  • C 2 2 0
  • D 0 1 2
  • E 1 1 4

Express 𝐴(𝐵+𝐶) in terms of 𝐴𝐵 and 𝐴𝐶.

  • A 𝐴 𝐵 + 𝐶
  • B 𝐵 𝐴 + 𝐶 𝐴
  • C 𝐵 𝐴 + 𝐶
  • D 𝐵 + 𝐴 𝐶
  • E 𝐴 𝐵 + 𝐴 𝐶

Q12:

Given that 𝐴=032161𝐵=5614𝐶=3042,,, is it true that (𝐴𝐵)𝐶=𝐴(𝐵𝐶)?

  • Ano
  • Byes

Q13:

From the following, choose two 2×2 matrices, 𝐴 and 𝐵, such that 𝐴0, 𝐵0, and 𝐴𝐵𝐵𝐴.

  • A 𝐴 = 1 2 3 4 , 𝐵 = 7 1 0 1 5 2 2
  • B 𝐴 = 1 0 0 4 , 𝐵 = 2 0 0 3
  • C 𝐴 = 1 2 3 4 , 𝐵 = 0 1 1 0
  • D 𝐴 = 1 1 1 1 , 𝐵 = 1 1 1 1
  • E 𝐴 = 1 2 3 4 , 𝐵 = 1 2 3 4

Q14:

Given that 𝐴=14111 and 𝐼 is the identity matrix of the same order as 𝐴, find 𝐴×𝐼 and 𝐼.

  • A 𝐴 × 𝐼 = 𝐴 , 𝐼 = 𝐼
  • B 𝐴 × 𝐼 = 𝐴 , 𝐼 = 𝑛 𝐼
  • C 𝐴 × 𝐼 = 𝐴 , 𝐼 = 𝑛 𝐼
  • D 𝐴 × 𝐼 = 𝐴 , 𝐼 = 𝐼

Q15:

From the following, choose two 2×2 matrices, 𝐴 and 𝐵, such that 𝐴0, 𝐵0 with 𝐴𝐵=0.

  • A 𝐴 = 1 1 1 1 , 𝐵 = 1 1 1 1
  • B 𝐴 = 1 0 0 4 , 𝐵 = 2 0 0 3
  • C 𝐴 = 1 2 3 4 , 𝐵 = 0 1 1 0
  • D 𝐴 = 1 1 1 1 , 𝐵 = 1 1 1 1
  • E 𝐴 = 1 2 3 4 , 𝐵 = 0 1 0 0

Q16:

If 𝐴 and 𝐵 are symmetric matrices, then the product 𝐴𝐵 is also symmetric only when 𝐴 and 𝐵 are .

  • Amatrices that commute
  • BHermitian matrices
  • Csquare matrices
  • Dinvertible matrices

Q17:

Let 𝐴=1230,𝐵=1022, and 𝐶=2104.

Find 𝐴𝐵.

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

Find (𝐴𝐵)𝐶.

  • A 0 8 1 0 1 3
  • B 3 3 1 1 0
  • C 3 3 1 1 0
  • D 1 0 2 1 6 3
  • E 1 0 2 1 6 3

Find 𝐵𝐶.

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

Find 𝐴(𝐵𝐶).

  • A 1 0 2 1 6 3
  • B 7 3 3 4
  • C 7 3 3 4
  • D 1 0 2 1 6 3
  • E 0 8 1 0 1 3

Q18:

What is the value of 𝐴+(𝐴) for any matrix 𝐴?

  • A 𝐴
  • B 𝑂
  • C 𝐴
  • D 1 0 0 1

Q19:

Let 𝑍 be a 2×3 matrix whose entries are all zero. If 𝐴 is any 2×3 matrix and 𝐵 is any 2×2 matrix, which of following is equivalent to 𝐴+𝐵𝑍?

  • A 𝐴
  • B 𝑍
  • C 𝐵
  • D 𝐴 + 𝐵
  • E 𝐴 𝐵 𝑍

Q20:

Given that 𝐴=543114𝐵=5231𝐶=0423,,, is it true that (𝐴𝐵)𝐶=𝐴(𝐵𝐶)?

  • Ano
  • Byes

Q21:

Let 𝐴=11102 and 𝐼 be the 2×2 identity matrix. Find 𝐴3𝐼, 𝐴+4𝐼, and their product (𝐴3𝐼)(𝐴+4𝐼), and then use this to express 𝐴 as a combination of 𝐴 and 𝐼.

  • A 0 1 1 0 3 , 5 1 1 0 2 , 1 0 2 2 0 4 , 𝐴 = ( 1 ) 𝐴 + 1 2 𝐼 .
  • B 2 1 1 0 5 , 5 1 1 0 2 , 1 2 7 7 0 2 , 𝐴 = 7 𝐴 + 1 2 𝐼 .
  • C 4 1 1 0 1 , 3 1 1 0 6 , 0 0 0 0 , 𝐴 = 𝐴 + 1 2 𝐼 .
  • D 2 1 1 0 5 , 5 1 1 0 2 , 0 0 0 0 , 𝐴 = ( 1 ) 𝐴 + 1 2 𝐼 .
  • E 4 1 1 0 1 , 3 1 1 0 6 , 1 1 1 1 0 1 4 , 𝐴 = 𝐴 + 1 2 𝐼 .

Q22:

If 𝐴=1342 and 𝐵=2011, is (7𝐴)𝐵=𝐴(7𝐵)?

  • Ayes
  • Bno

Q23:

Suppose that 𝐴=1342, 𝐵=2011, and 𝐶=0130.

Find 𝐴𝐵.

  • A 2 6 5 5
  • B 4 2 3 9
  • C 3 3 3 1
  • D 1 3 6 2
  • E 9 1 6 4

Find 𝐴𝐶.

  • A 1 3 6 2
  • B 3 3 3 1
  • C 9 1 6 4
  • D 1 2 7 2
  • E 2 6 5 5

Find 𝐴(2𝐵+7𝐶).

  • A 2 4 2 1 1 5 3
  • B 2 4 2 4
  • C 8 4 1 2 6
  • D 6 1 1 3 5 4 3 2
  • E 2 1 3 3 3 4

Express 𝐴(2𝐵+7𝐶) in terms of 𝐴𝐵 and 𝐴𝐶.

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

Q24:

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

  • ANo, they are different matrices of the same dimensions.
  • BNo, they have different dimensions.
  • CYes, they are both the 3×3 identity matrix.

Q25:

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

  • AYes, they are both the identity matrix.
  • BNo, they have different dimensions.
  • CNo, they are different matrices of the same dimensions.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.