Linear Algebra

JKPSC Assistant Professor (Mathematics) · Mathematics · 63 practice questions with answers and explanations: 38 Easy, 25 Medium, 0 Hard.

Sample questions

Sample Easy question 1

What is the rank of the matrix [2 1 1; 2 3 0; −2 −3 0]?

  1. 2
  2. 0
  3. 1
  4. 3
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Answer: A. Row reduction leaves 2 non-zero rows, so the rank is 2.

Sample Easy question 2

A linear map from R⁷ to R³ has rank 1. What is its nullity?

  1. 6
  2. 2
  3. 8
  4. 1
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Answer: A. By rank–nullity, nullity = 7 − 1 = 6.

Sample Easy question 3

The eigenvalues of the upper triangular matrix [5 1; 0 6] are:

  1. 5 and −6
  2. 5 and 6
  3. 30 and 1
  4. 11 and 0
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Answer: B. The eigenvalues of a triangular matrix are its diagonal entries.

Sample Medium question 4

A 3×3 matrix has eigenvalues −1, 8 and 4. What is its determinant?

  1. −32
  2. −4
  3. −31
  4. 11
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Answer: A. The determinant is the product of the eigenvalues: −32.

Sample Medium question 5

What is the rank of the matrix [−1 −1 −2; 2 2 4; 1 1 2]?

  1. 1
  2. 0
  3. 2
  4. 3
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Answer: A. Row reduction leaves 1 non-zero rows, so the rank is 1.

Sample Medium question 6

What is the determinant of the matrix [5 4 3; 1 −2 2; 1 −3 2]?

  1. −7
  2. 7
  3. 2
  4. 11
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Answer: B. Expanding along the first row gives 7.

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