Worksheet: Properties of Matrix Multiplication

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=4224,𝐵=3311, find 𝐴𝐵 and 𝐵𝐴.

  • A𝐴𝐵=1014210, 𝐵𝐴=6666
  • B𝐴𝐵=1021410, 𝐵𝐴=6666
  • C𝐴𝐵=1014210, 𝐵𝐴=1014210
  • D𝐴𝐵=6666, 𝐵𝐴=6666

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=8312 and 𝐵=8312, is 𝐴𝐵=𝐵𝐴?

  • Ano
  • Byes

Q6:

Given that 2×2 matrices 𝐴=1342 and 𝐵=1391216, 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=2105, 𝐵=01 and 𝐶=13.

Find 𝐴𝐵.

  • A25
  • B15
  • C05
  • D15
  • E25

Find 𝐴𝐶.

  • A215
  • B515
  • C215
  • D115
  • E216

Find 𝐴(𝐵+𝐶).

  • A14
  • B420
  • C220
  • D012
  • E114

Express 𝐴(𝐵+𝐶) in terms of 𝐴𝐵 and 𝐴𝐶.

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

Q12:

Given that 𝐴=032161𝐵=5614𝐶=3042,,, is it true that (𝐴𝐵)𝐶=𝐴(𝐵𝐶)?

  • Ano
  • Byes

Q13:

From the following, choose two 2×2 matrices, 𝐴 and 𝐵, such that 𝐴0, 𝐵0, and 𝐴𝐵𝐵𝐴.

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

Q14:

Given that 𝐴=14111 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 𝐴0, 𝐵0 with 𝐴𝐵=0.

  • A𝐴=1111, 𝐵=1111
  • B𝐴=1004, 𝐵=2003
  • C𝐴=1234, 𝐵=0110
  • D𝐴=1111, 𝐵=1111
  • 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=1230,𝐵=1022, and 𝐶=2104.

Find 𝐴𝐵.

  • A5430
  • B5430
  • C5430
  • D1284
  • E1284

Find (𝐴𝐵)𝐶.

  • A081013
  • B33110
  • C33110
  • D102163
  • E102163

Find 𝐵𝐶.

  • A21410
  • B21410
  • C4588
  • D4288
  • E4288

Find 𝐴(𝐵𝐶).

  • A102163
  • B7334
  • C7334
  • D102163
  • E081013

Q18:

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

  • A𝐴
  • 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=543114𝐵=5231𝐶=0423,,, is it true that (𝐴𝐵)𝐶=𝐴(𝐵𝐶)?

  • Ano
  • Byes

Q21:

Let 𝐴=11102 and 𝐼 be the 2×2 identity matrix. Find 𝐴3𝐼, 𝐴+4𝐼, and their product (𝐴3𝐼)(𝐴+4𝐼), and then use this to express 𝐴 as a combination of 𝐴 and 𝐼.

  • A01103, 51102, 102204, 𝐴=(1)𝐴+12𝐼.
  • B21105, 51102, 127702, 𝐴=7𝐴+12𝐼.
  • C41101, 31106, 0000, 𝐴=𝐴+12𝐼.
  • D21105, 51102, 0000, 𝐴=(1)𝐴+12𝐼.
  • E41101, 31106, 1111014, 𝐴=𝐴+12𝐼.

Q22:

If 𝐴=1342 and 𝐵=2011, is (7𝐴)𝐵=𝐴(7𝐵)?

  • Ayes
  • Bno

Q23:

Suppose that 𝐴=1342, 𝐵=2011, and 𝐶=0130.

Find 𝐴𝐵.

  • A2655
  • B4239
  • C3331
  • D1362
  • E9164

Find 𝐴𝐶.

  • A1362
  • B3331
  • C9164
  • D1272
  • E2655

Find 𝐴(2𝐵+7𝐶).

  • A2421153
  • B2424
  • C84126
  • D61135432
  • E213334

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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