Worksheet: Properties of Matrix Multiplication

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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=−422−4,𝐵=−3−3−11, find 𝐴𝐵 and 𝐵𝐴.

  • A𝐴𝐵=1014−2−10, 𝐵𝐴=666−6
  • B𝐴𝐵=10−214−10, 𝐵𝐴=666−6
  • C𝐴𝐵=1014−2−10, 𝐵𝐴=1014−2−10
  • D𝐴𝐵=666−6, 𝐵𝐴=666−6

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.

  • A𝐴(𝐵(𝐶𝐷))=(𝐴(𝐵𝐶))𝐷=𝐴((𝐵𝐶)𝐷)=((𝐴𝐵)𝐶)𝐷
  • B𝐴(𝐵(𝐶𝐷))=𝐴((𝐵𝐶)𝐷)=(𝐴(𝐵𝐶))𝐷=((𝐴𝐵)𝐶)𝐷
  • C𝐴(𝐵(𝐶𝐷))=𝐴((𝐵𝐶)𝐷)=(𝐴(𝐵+𝐶))𝐷=((𝐴+𝐵)𝐶)𝐷
  • D𝐴(𝐵(𝐶𝐷))=(𝐴(𝐵𝐶)𝐷)=((𝐴𝐵)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=8−31−2 and 𝐵=8−31−2, is 𝐴𝐵=𝐵𝐴?

  • Ano
  • Byes

Q6:

Given that 2×2 matrices 𝐴=1−3−42 and 𝐵=13−9−1216, 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)?

  • A𝐵𝐴+𝐶𝐴
  • B𝐴𝐵+𝐶
  • C𝐵+𝐴𝐶
  • D𝐵𝐴+𝐶
  • E𝐴𝐵+𝐴𝐶

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=210−5, 𝐵=0−1 and 𝐶=1−3.

Find 𝐴𝐵.

  • A2−5
  • B−15
  • C0−5
  • D−1−5
  • E−2−5

Find 𝐴𝐶.

  • A215
  • B5−15
  • C2−15
  • D−115
  • E216

Find 𝐴(𝐵+𝐶).

  • A1−4
  • B4−20
  • C−220
  • D012
  • E−114

Express 𝐴(𝐵+𝐶) in terms of 𝐴𝐵 and 𝐴𝐶.

  • A𝐴𝐵+𝐶
  • B𝐵𝐴+𝐶𝐴
  • C𝐵𝐴+𝐶
  • D𝐵+𝐴𝐶
  • E𝐴𝐵+𝐴𝐶

Q12:

Given that 𝐴=03−216−1𝐵=−5−614𝐶=−304−2,,, is it true that (𝐴𝐵)𝐶=𝐴(𝐵𝐶)?

  • Ano
  • Byes

Q13:

From the following, choose two 2×2 matrices, 𝐴 and 𝐵, such that 𝐴≠0, 𝐵≠0, and 𝐴𝐵≠𝐵𝐴.

  • A𝐴=1234, 𝐵=7101522
  • B𝐴=1004, 𝐵=−2003
  • C𝐴=1234, 𝐵=0110
  • D𝐴=1111, 𝐵=1−1−11
  • E𝐴=1234, 𝐵=1234

Q14:

Given that 𝐴=−14−111 and 𝐼 is the identity matrix of the same order as 𝐴, find 𝐴×𝐼 and 𝐼.

  • A𝐴×𝐼=𝐴, 𝐼=𝐼
  • B𝐴×𝐼=𝐴, 𝐼=𝑛𝐼
  • C𝐴×𝐼=𝐴, 𝐼=𝑛𝐼
  • D𝐴×𝐼=𝐴, 𝐼=𝐼

Q15:

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

  • A𝐴=1−1−11, 𝐵=1111
  • B𝐴=1004, 𝐵=−2003
  • C𝐴=1234, 𝐵=0110
  • D𝐴=1−111, 𝐵=1−1−11
  • E𝐴=1234, 𝐵=0100

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=1−230,𝐵=−1022, and 𝐶=−2104.

Find 𝐴𝐵.

  • A−5−4−30
  • B−5430
  • C54−30
  • D−128−4
  • E1284

Find (𝐴𝐵)𝐶.

  • A0−8−1013
  • B3−3−110
  • C33−110
  • D102163
  • E10−216−3

Find 𝐵𝐶.

  • A21410
  • B2−1−410
  • C4588
  • D4−2−88
  • E4288

Find 𝐴(𝐵𝐶).

  • A102163
  • B7334
  • C−7−3−34
  • D10−216−3
  • E0−8−1013

Q18:

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

  • A−𝐴
  • B𝑂
  • C𝐴
  • D1001

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?

  • A𝐴
  • B𝑍
  • C𝐵
  • D𝐴+𝐵
  • E𝐴𝐵𝑍

Q20:

Given that 𝐴=5−4−3−11−4𝐵=523−1𝐶=0−42−3,,, is it true that (𝐴𝐵)𝐶=𝐴(𝐵𝐶)?

  • Ano
  • Byes

Q21:

Let 𝐴=1110−2 and 𝐼 be the 2×2 identity matrix. Find 𝐴−3𝐼, 𝐴+4𝐼, and their product (𝐴−3𝐼)(𝐴+4𝐼), and then use this to express 𝐴 as a combination of 𝐴 and 𝐼.

  • A0110−3, 51102, 102204, 𝐴=(−1)𝐴+12𝐼.
  • B−2110−5, 51102, 12770−2, 𝐴=7𝐴+12𝐼.
  • C41101, −3110−6, 0000, 𝐴=𝐴+12𝐼.
  • D−2110−5, 51102, 0000, 𝐴=(−1)𝐴+12𝐼.
  • E41101, −3110−6, −11110−14, 𝐴=𝐴+12𝐼.

Q22:

If 𝐴=1−3−42 and 𝐵=201−1, is (7𝐴)𝐵=𝐴(7𝐵)?

  • Ayes
  • Bno

Q23:

Suppose that 𝐴=1−3−42, 𝐵=201−1, and 𝐶=01−30.

Find 𝐴𝐵.

  • A2−65−5
  • B−42−39
  • C3−3−31
  • D−13−6−2
  • E91−6−4

Find 𝐴𝐶.

  • A−13−6−2
  • B3−3−31
  • C91−6−4
  • D1−2−72
  • E2−65−5

Find 𝐴(2𝐵+7𝐶).

  • A−242−1153
  • B−2−424
  • C84−12−6
  • D6113−54−32
  • E−213−33−4

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

  • A2𝐵𝐴+7𝐶𝐴
  • B2𝐴𝐵+7𝐴𝐶
  • C2𝐴𝐵+7𝐶
  • D2𝐵+7𝐴𝐶
  • E2𝐵𝐴+7𝐶

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.

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