Worksheet: Inverse of a Matrix: Row Operations

In this worksheet, we will practice using elementary row operations to find the inverse of a matrix, if possible.

Q1:

Using elementary row operations, find 𝐴, if possible, for the matrix 𝐴=011511233.

  • A 𝐴 = 1 4 6 0 2 1 3 2 5 1 7 2 5
  • B 𝐴 = 1 4 6 0 2 1 3 2 5 1 7 2 5
  • C 𝐴 = 1 4 6 0 2 1 3 2 5 1 7 2 5
  • D 𝐴 = 1 4 6 0 2 1 3 2 5 1 7 2 5
  • E The matrix has no inverse.

Q2:

Find the multiplicative inverse of 1470010001.

  • A 1 4 7 0 0 1 0 0 0 1
  • B 1 0 0 4 7 1 0 0 0 1
  • C 1 4 7 0 0 1 0 0 0 1
  • D 1 0 0 4 7 1 0 0 0 1

Q3:

Use the inverse matrix to solve 3585123124721353𝑥𝑦𝑧𝑤=0101. Give your answer as an appropriate matrix.

  • A 7 1 8 8 1
  • B 7 8 2 2
  • C 1 . 7 5 4 . 5 2 0 . 2 5
  • D 2 5 2 0
  • E 7 8 2 2

Q4:

Find 𝐴, given that 𝐴=1202112021321212.

  • A 1 0 0 0 0 1 2 0 3 1 3 3 1 0 0 1
  • B 0 0 1 0 1 0 1 2 0 1 3 1 1 3 0 8 9 1 3 1 9 1
  • C 1 1 2 1 2 1 2 3 1 2 1 2 5 2 1 0 0 1 2 3 4 1 4 9 4
  • D 1 1 2 1 2 1 2 3 1 2 1 2 5 2 1 0 0 1 2 3 2 1 2 9 2
  • E 1 1 1 1 3 1 1 5 1 0 0 1 2 3 1 9

Q5:

Find the multiplicative inverse of the matrix 100710801.

  • A 1 0 0 7 1 0 8 0 1
  • B 1 7 8 0 1 0 0 0 1
  • C 1 0 0 7 1 0 8 0 1
  • D 1 7 8 0 1 0 0 0 1

Q6:

Using the elementary row operation, find 𝐴 for the matrix 𝐴=5321 if possible.

  • A 𝐴 = 3 1 5 2
  • B 𝐴 = 1 3 2 5
  • C 𝐴 = 3 1 5 2
  • D 𝐴 = 1 3 2 5
  • E 𝐴 = 1 3 2 5

Q7:

Using elementary row operations, find 𝐴 for the matrix 𝐴=112303330 if possible.

  • A 𝐴 = 1 3 9 6 3 6 4 3 9 5 3
  • B 𝐴 = 1 3 9 6 3 6 4 3 9 5 3
  • C 𝐴 = 9 6 3 6 4 3 9 5 3
  • D The matrix has no inverse.
  • E 𝐴 = 1 3 9 6 3 6 4 3 9 5 3

Q8:

Solve this system of equations using the inverse matrix 1121212312125210012341494𝑥𝑦𝑧𝑤=𝑎𝑏𝑐𝑑.

Give your solution as an appropriate matrix whose elements are expressed in terms of 𝑎, 𝑏, 𝑐, and 𝑑.

  • A 𝑎 + 𝑏 + 2 𝑐 + 𝑑 2 𝑎 + 𝑏 + 𝑐 + 2 𝑑 2 𝑏 3 𝑐 + 𝑑 2 𝑎 + 2 𝑐 + 2 𝑑
  • B 𝑎 + 2 𝑏 + 𝑐 + 2 𝑑 𝑎 + 𝑏 + 2 𝑐 + 𝑑 2 𝑎 + 𝑏 3 𝑐 + 2 𝑑 𝑎 + 2 𝑏 + 𝑐 + 2 𝑑
  • C 𝑎 + 𝑏 + 2 𝑐 + 𝑑 2 𝑎 + 𝑏 + 𝑐 + 2 𝑑 𝑎 + 2 𝑏 3 𝑐 + 𝑑 2 𝑎 + 𝑏 + 2 𝑐 + 2 𝑑
  • D 𝑎 + 2 𝑏 + 2 𝑑 𝑎 + 𝑏 + 2 𝑐 2 𝑎 + 𝑏 3 𝑐 + 2 𝑑 𝑎 + 2 𝑏 + 𝑐 + 2 𝑑
  • E 𝑎 + 2 𝑏 + 𝑐 + 𝑑 2 𝑎 + 𝑏 + 𝑐 + 2 𝑑 3 𝑏 + 2 𝑐 + 𝑑 2 𝑎 + 2 𝑏 + 2 𝑑

Q9:

Consider 𝐴=1245. Find the inverse of the matrix 𝐴 using elementary row operations.

  • A 5 3 2 3 4 3 1 3
  • B 5 3 2 3 4 3 1 3
  • C 1 3 2 3 4 3 5 3
  • D 1 1 2 1 4 1 5
  • E 1 1 2 1 4 1 5

Q10:

If a set of row operations (𝐸) are performed on a matrix 𝐴 until it is transformed into the identity matrix, what is the matrix resulting from performing the same row operations on the identity matrix?

  • A 𝐴
  • B 𝐼
  • C0
  • D 𝐴
  • E 𝐴

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