Worksheet: Rank of a Matrix

In this worksheet, we will practice finding the rank and nullity of a matrix.

Q1:

What is the rank of the matrix [00]?

Q2:

Find the rank of the matrix 224448.

Q3:

Find the rank of the matrix 100411210020.

Q4:

Find the rank of the matrix 120321210021.

Q5:

Find the rank of the matrix 0102010032605401120220214032.

Q6:

Find the rank of the matrix 01021220321216801150230217034.

Q7:

If 𝐴 is an 𝑛×𝑛 matrix of rank 𝑝<𝑛,then what is the dimension of the null-space of 𝐴.

  • A𝑛𝑝
  • B𝑛𝑝+1
  • CIt cannot be determined from the data.
  • D𝑝
  • E0

Q8:

Find the rank of the matrix 2220582121.

Q9:

Find the rank of the matrix 1054502520.

Q10:

What is the rank of the matrix 700?

Q11:

What is the rank of the matrix [800]?

Q12:

What value can 𝑘 not take if the rank of the matrix 𝐴=741522𝑘2491521 is 3?

Q13:

Find the rank of 0102112032615101120210214031.

Q14:

Find the rank of the matrix 1611141719243624.

Q15:

If the matrix 𝐴=691𝑘244151811 has rank 2, what is the value of 𝑘?

Q16:

What is the rank of the identity matrix, 𝐼?

Q17:

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

Q18:

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

Q19:

Determine the value of 𝑘 that makes the matrix 779𝑘61221 have the lowest rank possible.

Q20:

What is the rank of the identity matrix, 𝐼?

Q21:

Determine the rank of the matrix 𝐴=4732035𝑘639 for all real values of 𝑘.

  • AWhen 𝑘3, 𝑟𝑘(𝐴)=2, and when 𝑘=3, 𝑟𝑘(𝐴)=3.
  • BWhen 𝑘3, 𝑟𝑘(𝐴)=3, and when 𝑘=3, 𝑟𝑘(𝐴)=2.
  • CWhen 𝑘15, 𝑟𝑘(𝐴)=2, and when 𝑘=15, 𝑟𝑘(𝐴)=3.
  • DWhen 𝑘15, 𝑟𝑘(𝐴)=3, and when 𝑘=15, 𝑟𝑘(𝐴)=2.

Q22:

Find the rank of the augmented matrix of the following system of equations: 9𝑥3𝑦=2,45𝑥15𝑦=10.

Q23:

Find the rank of the augmented matrix of the following system of equations: 8𝑥+18𝑦=10,4𝑥+9𝑦=5,3𝑥+5𝑦=8.

Q24:

Determine the order of an augmented matrix that represents 𝑚 linear equations in 𝑛 variables.

  • A(𝑚+1)×𝑛
  • B(𝑚+1)×(𝑛+1)
  • C𝑚×𝑛
  • D𝑚×(𝑛+1)

Q25:

Suppose 𝐴,𝐵, and 𝐶 are invertible 𝑛×𝑛 matrices. Which of the following is false?

  • A𝐴𝐵𝐶 does not have full rank.
  • B𝐴𝐶𝐵 is invertible.
  • Cdet(𝐶𝐴𝐵)0
  • D(𝐴𝐵𝐶)=𝐶𝐵𝐴

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