Worksheet: Elementary Row Operations

In this worksheet, we will practice performing elementary row operations on a matrix and how to represent a system of linear equations as an augmented matrix.

Q1:

Suppose that matrix 𝐵 is obtained by performing a sequence of row operations on 𝐴. Can 𝐴 be obtained by performing row operations on 𝐵?

  • AThere is not enough information provided to determine this.
  • BIn special cases
  • CYes
  • DNo

Q2:

Consider the matrix 𝐴=2631. After obtaining the row-equivalent matrix ̃𝐴 by performing the following elementary row operations in order: 𝑟12𝑟, 𝑟𝑟, 𝑟𝑟2𝑟, and 𝑟𝑟𝑟, what is the determinant of 𝐴 in terms of the determinant of the row-equivalent matrix ̃𝐴?

  • A|𝐴|=12|̃𝐴|
  • B|𝐴|=2|̃𝐴|
  • C|𝐴|=1.25|̃𝐴|
  • D|𝐴|=2|̃𝐴|
  • E|𝐴|=12|̃𝐴|

Q3:

Consider the matrix 𝐴=152201120. After obtaining the row-equivalent matrix ̃𝐴 by performing the following elementary row operations in order: 𝑟3𝑟, 𝑟2𝑟, 𝑟𝑟+𝑟, 𝑟𝑟, and 𝑟𝑟2𝑟, what is the determinant of 𝐴 in terms of the determinant of the row-equivalent matrix ̃𝐴?

  • A|𝐴|=6|̃𝐴|
  • B|𝐴|=6|̃𝐴|
  • C|𝐴|=16|̃𝐴|
  • D|𝐴|=|̃𝐴|
  • E|𝐴|=16|̃𝐴|

Q4:

Consider the matrix 𝐴=136021142. Use elementary row operations to reduce the matrix into upper-triangular form.

  • Ã𝐴=13602100152
  • B̃𝐴=1360210092
  • C̃𝐴=136021007
  • D̃𝐴=136021004
  • Ẽ𝐴=136021002

Calculate the determinant of matrix 𝐴.

Q5:

Given that matrix 𝐴=1326, what is the resulting matrix from the elementary row operation 𝑟𝑟3𝑟 on matrix 𝐴?

  • A13618
  • B1313
  • C1313
  • D51526
  • E13515

Q6:

Are the matrices 𝐴=100011 and 𝐵=100111 row equivalent?

  • ANo
  • BYes

Q7:

Use elementary row operations to reduce the matrix 𝐴=2431 to upper-triangular form.

  • Ã𝐴=24015
  • B̃𝐴=612014
  • C̃𝐴=612015
  • D̃𝐴=24014
  • Ẽ𝐴=61207

Calculate the determinant of matrix 𝐴.

Q8:

If 𝐴 and 𝐵 are two 𝑚×𝑛 matrices having the same rank, can we conclude that they are row equivalent?

  • ANo
  • BYes

Q9:

If 𝐿, 𝑀, and 𝑃 are three 𝑚×𝑛 matrices such that 𝐿 is row equivalent to 𝑀 and 𝑀 is row equivalent to 𝑃, can we conclude that 𝐿 is row equivalent to 𝑃?

  • ANo
  • BYes

Q10:

Find the augmented matrix for the following system of equations:

𝑥+5𝑦=3,3𝑥+5𝑦=1.

  • A133551
  • B153351
  • C133511
  • D513531
  • E153351

Q11:

Write the augmented matrix for the following system of equations: 𝑥+𝑦𝑧=5,𝑦𝑧=2,𝑥+𝑦𝑧=2.

  • A111501121112
  • B101511121112
  • C111511121112
  • D111501121112
  • E111501121112

Q12:

Fill in the blank: For the system of equations defined by 2 equations of 3 variables, the size of the augmented matrix is .

  • A3×2
  • B2×3
  • C3×3
  • D2×2
  • E2×4

Q13:

Which of the following matrices is the matrix 𝐴=1036010520320241 reduced to upper triangular form using elementary row operations?

  • Ã𝐴=1036010500314000233
  • B̃𝐴=103601150031400073
  • C̃𝐴=103601250031400073
  • D̃𝐴=1036010500310000233
  • Ẽ𝐴=103601150031000073

Calculate the determinant of matrix 𝐴.

Q14:

Which of the following matrices is the matrix 𝐴=261131267 reduced to upper triangular form using elementary row operations?

  • Ã𝐴=131001005
  • B̃𝐴=131021005
  • C̃𝐴=131031006
  • D̃𝐴=131010.5006
  • Ẽ𝐴=130.5030.5008

Calculate the determinant of matrix 𝐴.

Q15:

𝐴 and 𝐵 are two 2×2 matrices defined as follows: 𝐴=1𝑏𝑐𝑑,𝐵=1001.

If 𝑑𝑐𝑏, is the matrix 𝐴 row equivalent to matrix 𝐵?

  • AYes
  • BNo

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