Worksheet: Rank of a Matrix by Determinant

In this worksheet, we will practice finding the rank of a given matrix by checking the greatest order determinant or minor determinant.

Q1:

If the matrix has rank 2, what is the value of 𝑘 ?

Q2:

Find the rank of the matrix

Q3:

What value can 𝑘 not take if the rank of the matrix is 3?

Q4:

What is the rank of the identity matrix, 𝐼 3 ?

Q5:

What is the rank of the zero matrix of order 3 × 3 ?

Q6:

What is the rank of the zero matrix of order 2 × 2 ?

Q7:

Determine the value of 𝑘 that makes the matrix have the lowest rank possible.

Q8:

What is the rank of the identity matrix, 𝐼 2 ?

Q9:

Determine the rank of the matrix 𝐴 = 4 7 3 2 0 3 5 𝑘 6 3 9 for all real values of 𝑘 .

  • AWhen 𝑘 3 , 𝑟 𝑘 ( 𝐴 ) = 3 , and when 𝑘 = 3 , 𝑟 𝑘 ( 𝐴 ) = 2 .
  • BWhen 𝑘 1 5 , 𝑟 𝑘 ( 𝐴 ) = 2 , and when 𝑘 = 1 5 , 𝑟 𝑘 ( 𝐴 ) = 3 .
  • CWhen 𝑘 3 , 𝑟 𝑘 ( 𝐴 ) = 2 , and when 𝑘 = 3 , 𝑟 𝑘 ( 𝐴 ) = 3 .
  • DWhen 𝑘 1 5 , 𝑟 𝑘 ( 𝐴 ) = 3 , and when 𝑘 = 1 5 , 𝑟 𝑘 ( 𝐴 ) = 2 .

Q10:

Find the rank of the following matrix.

  • A2
  • B3
  • C5
  • D4
  • E6

Q11:

What value can 𝑘 not take if the rank of the matrix is 3?

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