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=01151βˆ’12βˆ’3βˆ’3.

  • A𝐴=14602βˆ’132βˆ’517βˆ’25
  • B𝐴=14602βˆ’132βˆ’5βˆ’172βˆ’5
  • C𝐴=14602βˆ’132βˆ’517βˆ’25
  • D𝐴=1460213βˆ’2517βˆ’25
  • EThe matrix has no inverse.

Q2:

Find the multiplicative inverse of 1470010001.

  • A100βˆ’4710001
  • B1004710001
  • C1470010001
  • D1βˆ’470010001

Q3:

Use the inverse matrix to solve ⎑⎒⎒⎣3585123124721353⎀βŽ₯βŽ₯βŽ¦ο›π‘₯𝑦𝑧𝑀=⎑⎒⎒⎣0101⎀βŽ₯βŽ₯⎦. Give your answer as an appropriate matrix.

  • AβŽ‘βŽ’βŽ’βŽ£βˆ’782βˆ’2⎀βŽ₯βŽ₯⎦
  • BβŽ‘βŽ’βŽ’βŽ£βˆ’78βˆ’22⎀βŽ₯βŽ₯⎦
  • CβŽ‘βŽ’βŽ’βŽ£βˆ’1.754.5βˆ’2βˆ’0.25⎀βŽ₯βŽ₯⎦
  • DβŽ‘βŽ’βŽ’βŽ£βˆ’718βˆ’8βˆ’1⎀βŽ₯βŽ₯⎦
  • EβŽ‘βŽ’βŽ’βŽ£βˆ’25βˆ’20⎀βŽ₯βŽ₯⎦

Q4:

Find 𝐴, given that 𝐴=⎑⎒⎒⎣1202112021βˆ’321212⎀βŽ₯βŽ₯⎦.

  • A⎑⎒⎒⎣100001βˆ’203βˆ’13βˆ’3100βˆ’1⎀βŽ₯βŽ₯⎦
  • BβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’1121212312βˆ’12βˆ’52βˆ’1001βˆ’2βˆ’341494⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • CβŽ‘βŽ’βŽ’βŽ£βˆ’111131βˆ’1βˆ’5βˆ’1001βˆ’2βˆ’319⎀βŽ₯βŽ₯⎦
  • DβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’1121212312βˆ’12βˆ’52βˆ’1001βˆ’2βˆ’321292⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • E⎑⎒⎒⎒⎒⎒⎒⎣001010βˆ’120βˆ’131βˆ’130βˆ’89βˆ’13191⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦

Q5:

Find the multiplicative inverse of the matrix 100710801.

  • A1βˆ’7βˆ’8010001
  • B100710801
  • C100βˆ’710βˆ’801
  • D178010001

Q6:

Using the elementary row operation, find 𝐴 for the matrix 𝐴=5321 if possible.

  • A𝐴=ο”βˆ’132βˆ’5
  • B𝐴=1325
  • C𝐴=3152
  • D𝐴=ο”βˆ’1325
  • E𝐴=3βˆ’1βˆ’52

Q7:

Using elementary row operations, find 𝐴 for the matrix 𝐴=112303βˆ’330 if possible.

  • A𝐴=13ο˜βˆ’9636βˆ’4βˆ’39βˆ’5βˆ’3
  • B𝐴=13ο˜βˆ’963βˆ’643βˆ’953
  • CThe matrix has no inverse.
  • D𝐴=ο˜βˆ’963βˆ’6439βˆ’5βˆ’3
  • E𝐴=13ο˜βˆ’963βˆ’6439βˆ’5βˆ’3

Q8:

Solve this system of equations using the inverse matrix βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’1121212312βˆ’12βˆ’52βˆ’1001βˆ’2βˆ’341494⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦ο›π‘₯𝑦𝑧𝑀=βŽ‘βŽ’βŽ’βŽ£π‘Žπ‘π‘π‘‘βŽ€βŽ₯βŽ₯⎦.

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

  • AβŽ‘βŽ’βŽ’βŽ’βŽ£π‘Ž+2𝑏+𝑐+2π‘‘π‘Ž+𝑏+2𝑐+𝑑2π‘Ž+π‘βˆ’3𝑐+2π‘‘π‘Ž+2𝑏+𝑐+2π‘‘βŽ€βŽ₯βŽ₯βŽ₯⎦
  • BβŽ‘βŽ’βŽ’βŽ’βŽ£π‘Ž+2𝑏+2π‘‘π‘Ž+𝑏+2𝑐2π‘Ž+π‘βˆ’3𝑐+2π‘‘π‘Ž+2𝑏+𝑐+2π‘‘βŽ€βŽ₯βŽ₯βŽ₯⎦
  • CβŽ‘βŽ’βŽ’βŽ’βŽ£π‘Ž+2𝑏+𝑐+𝑑2π‘Ž+𝑏+𝑐+2π‘‘βˆ’3𝑏+2𝑐+𝑑2π‘Ž+2𝑏+2π‘‘βŽ€βŽ₯βŽ₯βŽ₯⎦
  • DβŽ‘βŽ’βŽ’βŽ’βŽ£π‘Ž+𝑏+2𝑐+𝑑2π‘Ž+𝑏+𝑐+2π‘‘π‘Ž+2π‘βˆ’3𝑐+𝑑2π‘Ž+𝑏+2𝑐+2π‘‘βŽ€βŽ₯βŽ₯βŽ₯⎦
  • EβŽ‘βŽ’βŽ’βŽ’βŽ£π‘Ž+𝑏+2𝑐+𝑑2π‘Ž+𝑏+𝑐+2𝑑2π‘βˆ’3𝑐+𝑑2π‘Ž+2𝑐+2π‘‘βŽ€βŽ₯βŽ₯βŽ₯⎦

Q9:

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

  • A⎑⎒⎒⎣1121415⎀βŽ₯βŽ₯⎦
  • BβŽ‘βŽ’βŽ’βŽ£βˆ’11214βˆ’15⎀βŽ₯βŽ₯⎦
  • CβŽ‘βŽ’βŽ’βŽ£βˆ’532343βˆ’13⎀βŽ₯βŽ₯⎦
  • DβŽ‘βŽ’βŽ’βŽ£βˆ’132343βˆ’53⎀βŽ₯βŽ₯⎦
  • E⎑⎒⎒⎣53βˆ’23βˆ’4313⎀βŽ₯βŽ₯⎦

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?

  • A0
  • B𝐼
  • C𝐴
  • D𝐴
  • E𝐴

Q11:

If 𝐼=πΈπΈβ€¦πΈπΈπ΄οŠοŠοŠ±οŠ§οŠ¨οŠ§, is 𝐴=πΌπΈπΈβ€¦πΈπΈοŠ±οŠ§οŠοŠοŠ±οŠ§οŠ¨οŠ§ true?

  • AYes
  • BNo

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