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Worksheet: Properties of Matrix Multiplication

Q1:

Given that find and .

  • A ,
  • B ,
  • C ,
  • D ,

Q2:

From the following, choose two 2 Γ— 2 matrices, 𝐴 and 𝐡 , such that 𝐴 β‰  0 , 𝐡 β‰  0 , and 𝐴 𝐡 β‰  𝐡 𝐴 .

  • A 𝐴 =  1 0 0 4  , 𝐡 =  βˆ’ 2 0 0 3 
  • B 𝐴 =  1 2 3 4  , 𝐡 =  1 2 3 4 
  • C 𝐴 =  1 1 1 1  , 𝐡 =  1 βˆ’ 1 βˆ’ 1 1 
  • D 𝐴 =  1 2 3 4  , 𝐡 =  0 1 1 0 
  • E 𝐴 =  1 2 3 4  , 𝐡 =  7 1 0 1 5 2 2 

Q3:

Are the matrices multiplicative inverses of each other?

  • AYes
  • BNo

Q4:

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
  • 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 = ( ( 𝐴 𝐡 ) 𝐢 ) 𝐷

Q5:

Consider the matrices By setting 𝐴 𝑋 βˆ’ 𝑋 𝐴 equal to the 3 Γ— 3 zero matrix, find matrices 𝐷 , 𝐸 , and 𝐹 such that if 𝐴 𝐡 = 𝐡 𝐴 , then 𝐡 = π‘Ž 𝐷 + 𝑏 𝐸 + β„Ž 𝐹 for some numbers π‘Ž , 𝑏 , and β„Ž .

  • A 𝐷 =  1 0 0 0 1 0 0 0 1  𝐸 =  0 βˆ’ 1 0 0 βˆ’ 1 0 0 0 βˆ’ 1  𝐹 =  0 0 0 0 0 0 0 βˆ’ 1 0  a n d a n d
  • B 𝐷 =  0 0 1 0 1 0 1 0 0  𝐸 =  0 1 0 0 βˆ’ 1 0 0 0 βˆ’ 1  𝐹 =  0 0 0 0 0 0 0 1 0  a n d a n d
  • C 𝐷 =  0 0 1 0 1 0 1 0 0  𝐸 =  0 βˆ’ 1 0 0 βˆ’ 1 0 0 0 βˆ’ 1  𝐹 =  0 0 0 0 0 0 0 βˆ’ 1 0  a n d a n d
  • D 𝐷 =  1 0 0 0 1 0 0 0 1  𝐸 =  0 1 0 0 βˆ’ 1 0 0 0 1  𝐹 =  0 0 0 0 0 0 0 1 0  a n d a n d
  • E 𝐷 =  0 0 1 0 1 0 1 0 0  𝐸 =  0 βˆ’ 1 0 0 1 0 0 0 1  𝐹 =  0 0 0 0 0 0 0 1 0  a n d a n d

Q6:

Given that is it true that ?

  • Ayes
  • Bno

Q7:

Given that is it true that ?

  • Ayes
  • Bno

Q8:

Consider the 2 Γ— 2 matrices 𝐴 =  1 1 0 0  and 𝐡 =  0 1 0 1  . Is 𝐴 𝐡 = 𝐡 𝐴 ?

  • Ano
  • Byes

Q9:

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

  • A yes
  • B no

Q10:

Given that 𝐴 =  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 = 𝑛 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 = 𝐼 2

Q11:

From the following, choose two 2 Γ— 2 matrices, 𝐴 and 𝐡 , such that 𝐴 β‰  0 , 𝐡 β‰  0 with 𝐴 𝐡 = 0 .

  • A 𝐴 =  1 0 0 4  , 𝐡 =  βˆ’ 2 0 0 3 
  • B 𝐴 =  1 2 3 4  , 𝐡 =  0 1 1 0 
  • C 𝐴 =  1 βˆ’ 1 1 1  , 𝐡 =  1 βˆ’ 1 βˆ’ 1 1 
  • D 𝐴 =  1 βˆ’ 1 βˆ’ 1 1  , 𝐡 =  1 1 1 1 
  • E 𝐴 =  1 2 3 4  , 𝐡 =  0 1 0 0 

Q12:

Given that 2 Γ— 2 matrices 𝐴 =  1 βˆ’ 3 βˆ’ 4 2  and 𝐡 =  1 3 βˆ’ 9 βˆ’ 1 2 1 6  , is 𝐴 𝐡 = 𝐡 𝐴 ?

  • Ayes
  • Bno

Q13:

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 .

  • Atrue
  • Bfalse

Q14:

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

Q15:

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

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

Q16:

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

  • Ayes
  • Bno

Q17:

What is the value of for any matrix ?

  • A
  • B
  • C
  • D

Q18:

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

Q19:

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 .

  • Afalse
  • Btrue

Q20:

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 ?

  • Ayes
  • Bno

Q21:

Find a matrix 𝐾 such that 𝐾 𝑋 = 𝑋 for all 2 Γ— 3 matrices 𝑋 .

  • A 𝐾 =  1 0 0 0 1 0 0 0 1 
  • B 𝐾 =  1 1 1 1 
  • C 𝐾 =  1 1 1 1 1 1 1 1 1 
  • D 𝐾 =  1 0 0 1 
  • E 𝐾 =  1 0 0 0 1 0 

Q22:

Suppose that 𝐴 =  2 1 0 βˆ’ 5  , 𝐡 =  0 βˆ’ 1  and 𝐢 =  1 βˆ’ 3  .

Find 𝐴 𝐡 .

  • A  0 βˆ’ 5 
  • B  βˆ’ 1 βˆ’ 5 
  • C  2 βˆ’ 5 
  • D  βˆ’ 1 5 
  • E  βˆ’ 2 βˆ’ 5 

Find 𝐴 𝐢 .

  • A  βˆ’ 1 1 5 
  • B  2 1 5 
  • C  2 1 6 
  • D  5 βˆ’ 1 5 
  • E  2 βˆ’ 1 5 

Find 𝐴 ( 𝐡 + 𝐢 ) .

  • A  1 βˆ’ 4 
  • B  4 βˆ’ 2 0 
  • C  βˆ’ 2 2 0 
  • D  0 1 2 
  • E  βˆ’ 1 1 4 

Express 𝐴 ( 𝐡 + 𝐢 ) in terms of 𝐴 𝐡 and 𝐴 𝐢 .

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

Q23:

Given the 2 Γ— 2 matrices 𝐴 =  8 βˆ’ 3 1 βˆ’ 2  and 𝐡 =  8 βˆ’ 3 1 βˆ’ 2  , is 𝐴 𝐡 = 𝐡 𝐴 ?

  • Ayes
  • Bno

Q24:

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

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

Q25:

Let 𝐴 =  1 βˆ’ 2 3 0  , 𝐡 =  βˆ’ 1 0 2 2  , and 𝐢 =  βˆ’ 2 1 0 4 . 

Find 𝐴 𝐡 .

  • A  βˆ’ 5 4 3 0 
  • B  βˆ’ 1 2 8 βˆ’ 4 
  • C  5 4 βˆ’ 3 0 
  • D  βˆ’ 5 βˆ’ 4 βˆ’ 3 0 
  • E  1 2 8 4 

Find ( 𝐴 𝐡 ) 𝐢 .

  • A  1 0 βˆ’ 2 1 6 βˆ’ 3 
  • B  3 βˆ’ 3 βˆ’ 1 1 0 
  • C  0 βˆ’ 8 βˆ’ 1 0 1 3 
  • D  1 0 2 1 6 3 
  • E  3 3 βˆ’ 1 1 0 

Find 𝐡 𝐢 .

  • A  4 5 8 8 
  • B  4 2 8 8 
  • C  2 βˆ’ 1 βˆ’ 4 1 0 
  • D  4 βˆ’ 2 βˆ’ 8 8 
  • E  2 1 4 1 0 

Find 𝐴 ( 𝐡 𝐢 ) .

  • A  1 0 βˆ’ 2 1 6 βˆ’ 3 
  • B  7 3 3 4 
  • C  βˆ’ 7 βˆ’ 3 βˆ’ 3 4 
  • D  1 0 2 1 6 3 
  • E  0 βˆ’ 8 βˆ’ 1 0 1 3 