Previous Year Questions on Rank of a Matrix

Rank of a Matrix - Previous Year Questions

Previous Year Questions on Rank of a Matrix

Previous year questions on Rank of a Matrix reveal the evolution and application of this concept in mathematical problem-solving. The concept of matrix rank emerged with the development of Linear Algebra Notes, focusing on matrix operations and their implications in theoretical and applied mathematics. It is essential in solving systems of linear equations and understanding vector spaces.

Vidyasagar University

2023-24 (CBCS)

Apply elementrary row operations on \(A\) to reduce it to a row echelon matrix. Hence, find the rank of \(A\) where \(A= \begin{pmatrix} 0 & 0 & 2 & 2 & 0\\ 1 & 3 & 2 & 4 & 1\\ 2 & 6 & 2 & 6 & 2\\ 3 & 9 & 1 & 10 & 6 \end{pmatrix}\). [5]

2021-22 (CBCS)

Find \(x\) if the rank of the matrix \(A= \begin{pmatrix} 1 & 3 & -3 & x \\ 2 & 2 & x & -4 \\ 1 & 1-x & 2x+1 & -8-3x \end{pmatrix}\) be \(2\). [4]

2020-21 (CBCS)

If the roots \(\alpha, \beta, \gamma \) of the equation \(x^{3}+qx+r=0 \) be in A.P. then show that the rank of the matrix \(\begin{pmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{pmatrix}\) is 2. [2]
Prove that interchange of two rows does not alter the rank of a matrix. [5]

2019-20 (CBCS)

Find all real \(\lambda \) for which the rank of the matrix \(A \) in \(2 \), where \(A= \begin{pmatrix} 1 & 2 & 3 & 1 \\ 2 & 5 & 3 & \lambda \\ 1 & 1 & 6 & \lambda+1 \end{pmatrix}\). [2]

2018-19 (CBCS)

Find a row echelon matrix which is row equivalent to \( \begin{pmatrix} 0 & 0 & 2 & 2 & 0\\ 1 & 3 & 2 & 4 & 1\\ 2 & 6 & 2 & 6 & 2\\ 3 & 9 & 1 & 10 & 6 \end{pmatrix}\). [5]
Obtain the fully row reduced normal form of the matrix \( \begin{pmatrix} 2 & 3 & -1 & -1 \\ 1 & -1 & -2 & -4 \\ 3 & 1 & 3 & -2 \\ 6 & 3 & 0 & -7 \end{pmatrix}\). [3]

2017-18 (CBCS)

Obtain the fully row reduced normal form of the matrix \( \begin{pmatrix} 0 & 0 & 1 & 2 & 1\\ 1 & 3 & 1 & 0 & 3\\ 2 & 6 & 4 & 2 & 8\\ 3 & 9 & 4 & 2 & 10 \end{pmatrix}\). [2]
Find the rank of the matrix \( \begin{pmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{pmatrix}\). If two straight lines \(a_{1}x+ b_{1}y+ c_{1}=0 \) and \(a_{2}x+ b_{2}y+ c_{2}=0 \) are coincident. [2]
Show that the rank of a skew symmetric matrix cannot be be \(1\). [2]
Find all real \(\lambda \) for which the rank of the matrix \(A \) in \(2 \), where \(A= \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & -1 & \lambda \\ 5 & 7 & 1 & \lambda^{2} \end{pmatrix}\). [3]

FAQs

What is the rank of a matrix?
It is the maximum number of linearly independent rows or columns in the matrix.
Why is the rank of a matrix important?
The rank determines the solutions of a system of linear equations and the dimensions of the column space.
What is the relationship between rank and determinant?
If a square matrix has a non-zero determinant, its rank equals its size.
What is the rank-nullity theorem?
It states that the sum of the rank and the nullity of a matrix equals the number of its columns.
Can the rank of a matrix be zero?
Yes, if all the elements of the matrix are zero, its rank is zero.
How is rank calculated?
Rank is calculated using row reduction to row echelon form or by evaluating the largest non-zero determinant.
What is the difference between full rank and rank-deficient matrices?
A full rank matrix has rank equal to its smaller dimension, while a rank-deficient matrix does not.
What are applications of rank in real life?
Applications include solving linear equations, network theory, and image compression techniques.
What is the rank of a diagonal matrix?
The rank of a diagonal matrix equals the number of its non-zero diagonal elements.
What role does rank play in eigenvalue problems?
The rank of a matrix helps determine the number of linearly independent eigenvectors.