Worksheet: Properties of Matrix Multiplication

In this worksheet, we will practice using the properties of matrix multiplication.

Q1:

Given that find 𝐴 𝐵 and 𝐵 𝐴 .

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

  • A 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 = ( ( 𝐴 𝐵 ) 𝐶 ) 𝐷

Q3:

Consider the matrices Is ?

  • Ano
  • Byes

Q4:

Are the following matrices multiplicative inverses of each other?

  • Ayes
  • Bno

Q5:

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

  • Ayes
  • Bno

Q6:

Given the matrices and , is ?

  • Ayes
  • Bno

Q7:

Given that matrices and , is ?

  • Ayes
  • Bno

Q8:

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

  • Afalse
  • Btrue

Q9:

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

  • Ayes
  • Bno

Q10:

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

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

Q11:

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 ) = 𝐴 𝐶 + 𝐴 𝐵 .

  • Atrue
  • Bfalse

Q12:

Suppose that , and .

Find .

  • A
  • B
  • C
  • D
  • E

Find .

  • A
  • B
  • C
  • D
  • E

Find .

  • A
  • B
  • C
  • D
  • E

Express in terms of and .

  • A
  • B
  • C
  • D
  • E

Q13:

Given that is it true that ( 𝐴 𝐵 ) 𝐶 = 𝐴 ( 𝐵 𝐶 ) ?

  • Ayes
  • Bno

Q14:

From the following, choose two matrices, and , such that , , and .

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

Q15:

Consider the matrices By setting equal to the zero matrix, find matrices and such that if , then for some numbers , and .

  • A
  • B
  • C
  • D
  • E

Q16:

Suppose 𝐴 𝐵 = 𝐴 𝐶 and 𝐴 is an invertible 𝑛 × 𝑛 matrix. Does it follow that 𝐵 = 𝐶 ?

  • A yes
  • B no

Q17:

Given that 𝐴 = 1 4 1 1 1 and 𝐼 is the identity matrix of the same order as 𝐴 , find 𝐴 × 𝐼 and 𝐼 2 .

  • A 𝐴 × 𝐼 = 𝐴 , 𝐼 = 𝑛 𝐼 2
  • B 𝐴 × 𝐼 = 𝐴 𝑇 , 𝐼 = 𝐼 2
  • C 𝐴 × 𝐼 = 𝐴 𝑇 , 𝐼 = 𝑛 𝐼 2
  • D 𝐴 × 𝐼 = 𝐴 , 𝐼 = 𝐼 2

Q18:

From the following, choose two matrices, and , such that , with .

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

Q19:

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

  • A Hermitian matrices
  • B square matrices
  • C invertible matrices
  • D matrices that commute

Q20:

Let , and

Find .

  • A
  • B
  • C
  • D
  • E

Find .

  • A
  • B
  • C
  • D
  • E

Find .

  • A
  • B
  • C
  • D
  • E

Find .

  • A
  • B
  • C
  • D
  • E

Q21:

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

  • A 𝐴
  • B 𝐴
  • C 1 0 0 1
  • D 𝑂

Q22:

If the matrices 𝐴 and 𝐵 both have order 𝑚 × 𝑛 , then what is the order of the matrix 𝐴 2 𝐵 ?

  • A 𝑚 × 1
  • B 𝑛 × 𝑚
  • C 1 × 𝑛
  • D 𝑚 × 𝑛

Q23:

Find a matrix such that for all matrices .

  • A
  • B
  • C
  • D
  • E

Q24:

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 + 𝐵 𝑍 ?

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

Q25:

Given that is it true that 𝐴 ( 𝐵 + 𝐶 ) = 𝐴 𝐵 + 𝐴 𝐶 ?

  • Ayes
  • Bno

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