Worksheet: Elementary Matrices

In this worksheet, we will practice identifying elementary matrices and their relation with row operations and how to find the inverse of an elementary matrix.

Q1:

Consider the matrix

𝐴=130329413.

Write the elementary matrix corresponding to the row operation 𝑟𝑟3𝑟.

  • A100310001
  • B300020001
  • C300010001
  • D100020001
  • E110310001

Derive the subsequent row-equivalent matrix ̃𝐴.

  • Ã𝐴=130679413
  • B̃𝐴=1300119413
  • C̃𝐴=13001112413
  • D̃𝐴=1306712413
  • Ẽ𝐴=130079413

Q2:

Consider the matrix

𝐴=3048211018320397.

Write the elementary matrix corresponding to the row operation 𝑟𝑟.

  • A1000000100100100
  • B0000000100000100
  • C1000010000100001
  • D1000111100101111
  • E1000000101000010

Derive the subsequent row-equivalent matrix ̃𝐴.

  • Ã𝐴=3048211018320397
  • B̃𝐴=3840201112380793
  • C̃𝐴=0000039700002110
  • D̃𝐴=3048271018320391
  • Ẽ𝐴=3048039718322110

Is it true that multiplying ̃𝐴 by the inverse elementary matrix on the left side will return the original matrix 𝐴?

  • ANo
  • BYes

Q3:

Consider the matrix

𝐴=122230413.

Write a single matrix 𝑀 corresponding to the combined row operations 𝑟2𝑟, 𝑟𝑟3𝑟, 𝑟𝑟+5𝑟, 𝑟𝑟, and 𝑟𝑟𝑟, in this order.

  • A𝑀=010200651
  • B𝑀=100200651
  • C𝑀=010210651
  • D𝑀=100210351
  • E𝑀=010210351

Use 𝑀 to calculate ̃𝐴.

  • Ã𝐴=2300141029
  • B̃𝐴=230014029
  • C̃𝐴=2300140210
  • D̃𝐴=23001410310
  • Ẽ𝐴=230014039

Q4:

Consider the matrix

𝐴=414304221098.

Write the elementary matrix corresponding to the row operation 𝑟𝑟+12𝑟.

  • A1120010001
  • B0320010001
  • C3200010001
  • D1000120001
  • E1200010001

Derive the subsequent row-equivalent matrix ̃𝐴.

  • Ã𝐴=433402121098
  • B̃𝐴=415202121098
  • C̃𝐴=415204221098
  • D̃𝐴=690704221098
  • Ẽ𝐴=433404221098

Is it true that multiplying ̃𝐴 by the inverse elementary matrix on the left side will return the original matrix 𝐴?

  • AYes
  • BNo

Q5:

Consider the matrix

𝐴=205135012110.

Write the elementary matrix corresponding to the row operation 𝑟𝑟.

  • A111010111
  • B001000100
  • C100010001
  • D100000101
  • E001010100

Derive the subsequent row-equivalent matrix ̃𝐴.

  • Ã𝐴=350120512110
  • B̃𝐴=150235010112
  • C̃𝐴=105215030112
  • D̃𝐴=211035012051
  • Ẽ𝐴=205121103501

Q6:

Consider the matrix 𝐴=2081115124080310.

Write the elementary matrix corresponding to the row operation 𝑟12𝑟.

  • A12000012000012000012
  • B1000010012121120001
  • C10000100001200001
  • D10000100001200001
  • E10000100121212120001

Derive the subsequent row-equivalent matrix ̃𝐴.

  • Ã𝐴=2081115112040310
  • B̃𝐴=20811151240.580310
  • C̃𝐴=20811151480160310
  • D̃𝐴=2081115112040310
  • Ẽ𝐴=20811151480160310

Q7:

Consider the matrix 𝐴=310124.

Write the elementary matrix corresponding to the row operation 𝑟2𝑟.

  • A1022
  • B1002
  • C2002
  • D1002

Derive the subsequent row-equivalent matrix ̃𝐴.

  • Ã𝐴=620248
  • B̃𝐴=310248
  • C̃𝐴=310144
  • D̃𝐴=310248
  • Ẽ𝐴=620248

Is it true that multiplying ̃𝐴 by the inverse of the elementary matrix on the left side will return the original matrix 𝐴?

  • ANo
  • BYes

Q8:

State whether the following statements are true or false.

Every elementary matrix is invertible.

  • ATrue
  • BFalse

The product of elementary matrices is invertible.

  • ATrue
  • BFalse

Q9:

If 𝐸 is an elementary matrix that represents row operations to convert matrix 𝐴 to matrix 𝐵, then which of the following statements is true?

  1. 𝐴×𝐸=𝐵
  2. 𝐸×𝐵=𝐴
  3. 𝐸×𝐵=𝐴
  4. 𝐸×𝐴=𝐵
  • AAll of them
  • BI only
  • CIII and IV
  • DIII only
  • ENone of them

Q10:

If 𝐸×𝐴=𝐵, where 𝐸 is a single row exchange elementary matrix, then which of the following statements are true?

  1. |𝐴|=|𝐵|
  2. 𝐸×𝐵=𝐴
  3. 𝐴 and 𝐵 have the same eigenvalues.
  4. |𝐸|=0
  • AI, II, and III
  • BNone of the statements
  • CAll the statements
  • DI and II
  • EII and IV

Q11:

What is the elementary matrix describing the row operation 𝑟𝑟 for a 2×2 matrix?

  • A1001
  • B010100001
  • C0111
  • D0110
  • E1110

Q12:

What is the elementary matrix describing the row operation 𝑟=3𝑟 for a 2×2 matrix?

  • A1003
  • B3001
  • C3301
  • D3303
  • E0301

Q13:

What is the elementary matrix describing the row operation 𝑟=𝑟+2𝑟 for a 2×2 matrix?

  • A1011
  • B1201
  • C1211
  • D1102
  • E1112

Q14:

Which of the following is not an elementary matrix?

  • A1003
  • B1501
  • C0110
  • D1040

Q15:

Which of the following matrices is not an elementary matrix?

  • A5001
  • B1051
  • C3005
  • D3001
  • E1301

Q16:

Find the inverse of the elementary matrix 3001 using inspection.

  • A10013
  • B130013
  • C30013
  • D1003
  • E13001

Q17:

Find the inverse of the elementary matrix 120010001 by inspection.

  • A121001200012
  • B120010001
  • C121001200012
  • D120010001
  • E120010001

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