Lesson Worksheet: Elementary Matrices Mathematics

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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=⎛⎜⎜⎝30−482110−183203−97⎞⎟⎟⎠.

Write the elementary matrix corresponding to the row operation 𝑟↔𝑟.

  • A⎛⎜⎜⎝1000000100100100⎞⎟⎟⎠
  • B⎛⎜⎜⎝0000000100000100⎞⎟⎟⎠
  • C⎛⎜⎜⎝1000010000100001⎞⎟⎟⎠
  • D⎛⎜⎜⎝1000111100101111⎞⎟⎟⎠
  • E⎛⎜⎜⎝1000000101000010⎞⎟⎟⎠

Derive the subsequent row-equivalent matrix ̃𝐴.

  • Ã𝐴=⎛⎜⎜⎝30−482110−183203−97⎞⎟⎟⎠
  • B̃𝐴=⎛⎜⎜⎝38−402011−123807−93⎞⎟⎟⎠
  • C̃𝐴=⎛⎜⎜⎝000003−9700002110⎞⎟⎟⎠
  • D̃𝐴=⎛⎜⎜⎝30−482710−183203−91⎞⎟⎟⎠
  • Ẽ𝐴=⎛⎜⎜⎝30−4803−97−18322110⎞⎟⎟⎠

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

  • AYes
  • BNo

Q2:

Consider the matrix

𝐴=13032−9413.

Write the elementary matrix corresponding to the row operation 𝑟→𝑟−3𝑟.

  • A100−310001
  • B−3000−20001
  • C−300010001
  • D1000−20001
  • E110−310001

Derive the subsequent row-equivalent matrix ̃𝐴.

  • Ã𝐴=1306−7−9413
  • B̃𝐴=1300−11−9413
  • C̃𝐴=1300−11−12413
  • D̃𝐴=1306−7−12413
  • Ẽ𝐴=1300−7−9413

Q3:

Consider the matrix

𝐴=41−4304221098.

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

  • A⎡⎢⎢⎣1120010001⎤⎥⎥⎦
  • B⎡⎢⎢⎣0320010001⎤⎥⎥⎦
  • C⎡⎢⎢⎣3200010001⎤⎥⎥⎦
  • D⎡⎢⎢⎣1000120001⎤⎥⎥⎦
  • E⎡⎢⎢⎣1200010001⎤⎥⎥⎦

Derive the subsequent row-equivalent matrix ̃𝐴.

  • Ã𝐴=43−3402121098
  • B̃𝐴=4−1−5202121098
  • C̃𝐴=4−1−5204221098
  • D̃𝐴=690704221098
  • Ẽ𝐴=43−3404221098

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

  • AYes
  • BNo

Q4:

Consider the matrix

𝐴=1−222−30−413.

Write a single matrix 𝑀 corresponding to the combined row operations 𝑟→2𝑟, 𝑟→𝑟−3𝑟, 𝑟→𝑟+5𝑟, 𝑟↔𝑟, and 𝑟→𝑟−𝑟, in this order.

  • A𝑀=010200−651
  • B𝑀=100200−651
  • C𝑀=0102−10−651
  • D𝑀=1002−10−351
  • E𝑀=0102−10−351

Use 𝑀 to calculate ̃𝐴.

  • Ã𝐴=2−300−1410−2−9
  • B̃𝐴=2−300−140−2−9
  • C̃𝐴=2−300−140−2−10
  • D̃𝐴=2−300−1410−3−10
  • Ẽ𝐴=2−300−140−3−9

Q5:

Consider the matrix

𝐴=2051350−12−110.

Write the elementary matrix corresponding to the row operation 𝑟↔𝑟.

  • A111010111
  • B001000100
  • C100010001
  • D100000101
  • E001010100

Derive the subsequent row-equivalent matrix ̃𝐴.

  • Ã𝐴=350−120512−110
  • B̃𝐴=1502350−101−12
  • C̃𝐴=1052−15030−112
  • D̃𝐴=2−110350−12051
  • Ẽ𝐴=20512−110350−1

Q6:

Consider the matrix 𝐴=⎡⎢⎢⎣2081−11512−4080310⎤⎥⎥⎦.

Write the elementary matrix corresponding to the row operation 𝑟→−12𝑟.

  • A⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−120000−120000−120000−12⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
  • B⎡⎢⎢⎢⎢⎣10000100−12−121−120001⎤⎥⎥⎥⎥⎦
  • C⎡⎢⎢⎢⎢⎣1000010000−1200001⎤⎥⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎢⎣10000100001200001⎤⎥⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎢⎣10000100−12−12−12−120001⎤⎥⎥⎥⎥⎦

Derive the subsequent row-equivalent matrix ̃𝐴.

  • Ã𝐴=⎡⎢⎢⎣2081115112040310⎤⎥⎥⎦
  • B̃𝐴=⎡⎢⎢⎣2081−11512−40.580310⎤⎥⎥⎦
  • C̃𝐴=⎡⎢⎢⎣2081−1151−480−160310⎤⎥⎥⎦
  • D̃𝐴=⎡⎢⎢⎣2081−1151−120−40310⎤⎥⎥⎦
  • Ẽ𝐴=⎡⎢⎢⎣2081−11514−80160310⎤⎥⎥⎦

Q7:

Consider the matrix 𝐴=310−124.

Write the elementary matrix corresponding to the row operation 𝑟→−2𝑟.

  • A10−2−2
  • B1002
  • C−200−2
  • D100−2

Derive the subsequent row-equivalent matrix ̃𝐴.

  • Ã𝐴=−6−20−248
  • B̃𝐴=3102−4−8
  • C̃𝐴=31014−4
  • D̃𝐴=310−248
  • Ẽ𝐴=6202−4−8

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

  • AYes
  • BNo

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:

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

  • A1003
  • B3001
  • C3301
  • D3303
  • E0301

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