Lesson Worksheet: Properties of Matrix Multiplication Mathematics

In this worksheet, we will practice identifying the properties of matrix multiplication, including the transpose of the product of two matrices, and comparing them to the properties of multiplication of numbers.


Given that 𝐴=ο”βˆ’422βˆ’4,𝐡=ο”βˆ’3βˆ’3βˆ’11, find 𝐴𝐡 and 𝐡𝐴.

  • A𝐴𝐡=1014βˆ’2βˆ’10, 𝐡𝐴=666βˆ’6
  • B𝐴𝐡=10βˆ’214βˆ’10, 𝐡𝐴=666βˆ’6
  • C𝐴𝐡=1014βˆ’2βˆ’10, 𝐡𝐴=1014βˆ’2βˆ’10
  • D𝐴𝐡=666βˆ’6, 𝐡𝐴=666βˆ’6


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(𝐢𝐷))=((𝐴𝐡)𝐢)𝐷.

  • A𝐴(𝐡(𝐢𝐷))=(𝐴(𝐡𝐢))𝐷=𝐴((𝐡𝐢)𝐷)=((𝐴𝐡)𝐢)𝐷
  • B𝐴(𝐡(𝐢𝐷))=𝐴((𝐡𝐢)𝐷)=(𝐴(𝐡𝐢))𝐷=((𝐴𝐡)𝐢)𝐷
  • C𝐴(𝐡(𝐢𝐷))=𝐴((𝐡𝐢)𝐷)=(𝐴(𝐡+𝐢))𝐷=((𝐴+𝐡)𝐢)𝐷
  • D𝐴(𝐡(𝐢𝐷))=(𝐴(𝐡𝐢)𝐷)=((𝐴𝐡)𝐢)𝐷


Consider the 2Γ—2 matrices 𝐴=1100 and 𝐡=0101. Is 𝐴𝐡=𝐡𝐴?

  • ANo
  • BYes


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

  • AYes
  • BNo


Given the 2Γ—2 matrices 𝐴=8βˆ’31βˆ’2 and 𝐡=8βˆ’31βˆ’2, is 𝐴𝐡=𝐡𝐴?

  • ANo
  • BYes


Given that 2Γ—2 matrices 𝐴=1βˆ’3βˆ’42 and 𝐡=13βˆ’9βˆ’1216, is 𝐴𝐡=𝐡𝐴?

  • AYes
  • BNo


True or False: If 𝐴 and 𝐡 are both 2Γ—2 matrices, then 𝐴𝐡 is never the same as 𝐡𝐴.

  • AFalse
  • BTrue


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

  • AYes
  • BNo


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

  • A𝐡𝐴+𝐢𝐴
  • B𝐴𝐡+𝐢
  • C𝐡+𝐴𝐢
  • D𝐡𝐴+𝐢
  • E𝐴𝐡+𝐴𝐢


True or False: If 𝐴 is a 2Γ—3 matrix and 𝐡 and 𝐢 are 3Γ—2 matrices, then 𝐴(𝐡+𝐢)=𝐴𝐡+𝐴𝐢.

  • ATrue
  • BFalse

This lesson includes 26 additional questions and 80 additional question variations for subscribers.

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