Worksheet: Gaussian Elimination

In this worksheet, we will practice using Gaussian elimination to get a row echelon form of a matrix and hence solve a system of linear equations.

Q1:

True or False: If a 4Γ—4 matrix is in row-echelon form, then the entry in the third row, second column must be 0.

  • AFalse
  • BTrue

Q2:

Using the row echelon form, check the number of solutions that the following system of linear equations has: π‘₯+𝑦+𝑧=6,2π‘₯βˆ’π‘¦+𝑧=3,2π‘₯+2𝑦+2𝑧=12.(2)(3)

  • AThe system has a unique solution.
  • BThe system has no solutions.
  • CThe system has an infinite number of solutions.

Q3:

Which of the following is the matrix 51532 in row echelon form?

  • A5150βˆ’7
  • B51500
  • C515730
  • D5151βˆ’7
  • E01500

Q4:

Find the row echelon form of the following matrix: ⎑⎒⎒⎣1111111101230123⎀βŽ₯βŽ₯⎦.

  • A⎑⎒⎒⎣1111012300110001⎀βŽ₯βŽ₯⎦
  • B⎑⎒⎒⎣1111012300230003⎀βŽ₯βŽ₯⎦
  • C⎑⎒⎒⎣1111012300000000⎀βŽ₯βŽ₯⎦
  • D⎑⎒⎒⎣1000012300230003⎀βŽ₯βŽ₯⎦
  • E⎑⎒⎒⎣1000012300000000⎀βŽ₯βŽ₯⎦

Q5:

The equation of a circle has the form π‘Žπ‘₯+π‘Žπ‘¦+𝑏π‘₯+𝑐𝑦+𝑑=0. Hence, a circle is determined by three points. Find the equation of the circle through the points (1,1), (2,1), and (3,2) by solving the equation using Gaussian elimination. Take 𝑑=6.

  • A2π‘₯+2𝑦+3π‘₯+5𝑦+6=0
  • Bπ‘₯+π‘¦βˆ’3π‘₯+5𝑦+6=0
  • Cπ‘₯+π‘¦βˆ’3π‘₯βˆ’5𝑦+6=0
  • D2π‘₯+2π‘¦βˆ’3π‘₯βˆ’5𝑦+6=0
  • Eπ‘₯+𝑦+3π‘₯+5𝑦+6=0

Q6:

Find the general solution for the system π‘₯βˆ’π‘¦+2𝑧=βˆ’3,4π‘₯+4π‘¦βˆ’2𝑧=1,βˆ’2π‘₯+2π‘¦βˆ’4𝑧=6.(2)(3)

  • Aο€Όβˆ’54𝑑+78,βˆ’54π‘‘βˆ’138,π‘‘οˆ
  • Bο€Ό34𝑑+118,34𝑑+58,π‘‘οˆ
  • Cο€Όβˆ’34π‘‘βˆ’118,54𝑑+138,π‘‘οˆ
  • Dο€Όβˆ’54π‘‘βˆ’78,βˆ’34π‘‘βˆ’58,π‘‘οˆ
  • Eο€Ό54π‘‘βˆ’78,βˆ’34𝑑+58,π‘‘οˆ

Q7:

Find the general solution for the system 2π‘₯+2𝑦+𝑧=0,2π‘₯βˆ’3π‘¦βˆ’4𝑧=0,4π‘₯βˆ’π‘¦βˆ’3𝑧=0.(2)(3)

  • Aο€Ό12𝑑,βˆ’π‘‘,π‘‘οˆ
  • B(2𝑑,βˆ’π‘‘,𝑑)
  • C(βˆ’2,1,1)
  • Dο€Όβˆ’12𝑑,βˆ’π‘‘,π‘‘οˆ
  • Eο€Όβˆ’12𝑑,𝑑,π‘‘οˆ

Q8:

Using Gaussian elimination, solve the following system of linear equations:

βˆ’3π‘₯+2π‘¦βˆ’6𝑧=6,5π‘₯+7π‘¦βˆ’5𝑧=6,π‘₯+4π‘¦βˆ’2𝑧=8.

  • Aπ‘₯=1,𝑦=3,𝑧=βˆ’2
  • Bπ‘₯=βˆ’1,𝑦=3,𝑧=2
  • Cπ‘₯=βˆ’1,𝑦=2,𝑧=3
  • Dπ‘₯=βˆ’2,𝑦=3,𝑧=1
  • Eπ‘₯=2,𝑦=3,𝑧=βˆ’1

Q9:

Which of the following is the matrix βŽ‘βŽ’βŽ’βŽ£βˆ’2βˆ’2βˆ’3βˆ’3βˆ’3201⎀βŽ₯βŽ₯⎦ in row echelon form?

  • AβŽ‘βŽ’βŽ’βŽ£βˆ’2βˆ’2010100⎀βŽ₯βŽ₯⎦
  • BβŽ‘βŽ’βŽ’βŽ£βˆ’3βˆ’3000000⎀βŽ₯βŽ₯⎦
  • CβŽ‘βŽ’βŽ’βŽ£βˆ’3βˆ’3010000⎀βŽ₯βŽ₯⎦
  • DβŽ‘βŽ’βŽ’βŽ£βˆ’2βˆ’2100100⎀βŽ₯βŽ₯⎦
  • EβŽ‘βŽ’βŽ’βŽ£βˆ’2βˆ’2110100⎀βŽ₯βŽ₯⎦

Q10:

Which of the following is the matrix ο˜βˆ’2βˆ’2231βˆ’3βˆ’213 in row echelon form?

  • Aο˜βˆ’2βˆ’220βˆ’20001
  • Bο˜βˆ’2βˆ’220βˆ’20000
  • Cο˜βˆ’2βˆ’22βˆ’200010
  • Dο˜βˆ’2βˆ’220βˆ’20010
  • Eο˜βˆ’2βˆ’22001000

Q11:

Using Gaussian elimination, solve the following system of linear equations: 6π‘₯+5𝑦=2,3π‘₯+7𝑦=4.

  • Aπ‘₯=βˆ’19,𝑦=13
  • Bπ‘₯=βˆ’29,𝑦=23
  • Cπ‘₯=βˆ’29,𝑦=βˆ’23
  • Dπ‘₯=19,𝑦=13
  • Eπ‘₯=βˆ’19,𝑦=23

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