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

Lesson: Properties of Matrix Multiplication

Sample Question Videos

Worksheet • 25 Questions • 1 Video

Q1:

Given that find 𝐴 𝐡 and 𝐡 𝐴 .

  • A 𝐴 𝐡 = ο€Ό 1 0 1 4 βˆ’ 2 βˆ’ 1 0  , 𝐡 𝐴 = ο€Ό 6 6 6 βˆ’ 6 
  • B 𝐴 𝐡 = ο€Ό 1 0 βˆ’ 2 1 4 βˆ’ 1 0  , 𝐡 𝐴 = ο€Ό 6 6 6 βˆ’ 6 
  • C 𝐴 𝐡 = ο€Ό 1 0 1 4 βˆ’ 2 βˆ’ 1 0  , 𝐡 𝐴 = ο€Ό 1 0 1 4 βˆ’ 2 βˆ’ 1 0 
  • D 𝐴 𝐡 = ο€Ό 6 6 6 βˆ’ 6  , 𝐡 𝐴 = ο€Ό 6 6 6 βˆ’ 6 

Q2:

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

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

Q3:

Consider the matrices Is ?

  • Ayes
  • Bno

Q4:

Are the following matrices multiplicative inverses of each other?

  • Ano
  • Byes

Q5:

Given the 1 Γ— 1 matrices 𝐴 = [ 3 ] and 𝐡 = [ 4 ] , is 𝐴 𝐡 = 𝐡 𝐴 ?

  • Ano
  • Byes

Q6:

Given the matrices and , is ?

  • Ano
  • Byes

Q7:

Given that matrices and , is ?

  • Ano
  • Byes

Q8:

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

  • Atrue
  • Bfalse

Q9:

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

  • Ano
  • Byes

Q10:

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

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

Q11:

State whether the following statement is true or false: If 𝐴 is a 2 Γ— 3 matrix and 𝐡 and 𝐢 are 3 Γ— 2 matrices, then 𝐴 ( 𝐡 + 𝐢 ) = 𝐴 𝐢 + 𝐴 𝐡 .

  • Afalse
  • Btrue

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 𝐡 ) 𝐢 = 𝐴 ( 𝐡 𝐢 ) ?

  • Ano
  • Byes

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 𝐡 = 𝐴 𝐢 and 𝐴 is an invertible 𝑛 Γ— 𝑛 matrix. Does it follow that 𝐡 = 𝐢 ?

  • A no
  • B yes

Q17:

Given that 𝐴 = ο€Ό βˆ’ 1 4 βˆ’ 1 1 1  and 𝐼 is the identity matrix of the same order as 𝐴 , find 𝐴 Γ— 𝐼 and 𝐼 2 .

  • A 𝐴 Γ— 𝐼 = 𝐴 , 𝐼 = 𝐼 2
  • B 𝐴 Γ— 𝐼 = 𝐴 𝑇 , 𝐼 = 𝐼 2
  • C 𝐴 Γ— 𝐼 = 𝐴 𝑇 , 𝐼 = 𝑛 𝐼 2
  • D 𝐴 Γ— 𝐼 = 𝐴 , 𝐼 = 𝑛 𝐼 2

Q18:

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

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

Q19:

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

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

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 + ( βˆ’ 𝐴 ) for any matrix 𝐴 ?

  • A 𝑂
  • B 𝐴
  • C ο€Ό 1 0 0 1 
  • D βˆ’ 𝐴

Q22:

If the matrices 𝐴 and 𝐡 both have order π‘š Γ— 𝑛 , then what is the order of the matrix 𝐴 βˆ’ 2 𝐡 ?

  • A π‘š Γ— 𝑛
  • B 𝑛 Γ— π‘š
  • C 1 Γ— 𝑛
  • D π‘š Γ— 1

Q23:

Find a matrix such that for all matrices .

  • A
  • B
  • C
  • D
  • E

Q24:

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

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

Q25:

Given that is it true that 𝐴 ( 𝐡 + 𝐢 ) = 𝐴 𝐡 + 𝐴 𝐢 ?

  • Ano
  • Byes
Preview