GATE Questions & Answers of Linear Algebra Computer Science and Information Technology

Let X be a square matrix. Consider the following two statements on X.

      I.           X is invertible.

      II.          Determinant of X is non-zero.

Which one of the following is TRUE?

Consider the following matrix:

$ R=\begin{bmatrix}1&2&4&8\\1&3&9&27\\1&4&16&64\\1&5&25&125\end{bmatrix} $

The absolute value of the product of Eigen values of R is .

Consider a matrix $\style{font-family:'Times New Roman'}{A=uv^T\;}$ where $\style{font-family:'Times New Roman'}{u=\begin{pmatrix}1\\2\end{pmatrix},\;v\;=\begin{pmatrix}1\\1\end{pmatrix}}$ . Note that $\style{font-family:'Times New Roman'}{v^T}$ denotes the transpose of $ v $. The largest eigenvalue of $ A $ is _____.

Consider a matrix P whose only eigenvectors are the multiples of $\style{font-family:'Times New Roman'}{\begin{bmatrix}1\\4\end{bmatrix}}$.

Consider the following statements.
(I) P does not have an inverse
(II) P has a repeated eigenvalue
(III) P cannot be diagonalized
Which one of the following options is correct?

Let N be the set of natural numbers. Consider the following sets.
P: Set of Rational numbers (positive and negative)
Q: Set of functions from {0, 1} to N
R: Set of functions from N to {0, 1}
S: Set of finite subsets of N.
Which of the sets above are countable?

Let c1,....,cn be scalars, not all zero, such that $\style{font-family:'Times New Roman'}{\sum_\limits{i=1}^nc_ia_i=0}$ where ai are column vectors in Rn Consider the set of linear equations
                                                      $\style{font-family:'Times New Roman'}{Ax=b}$
where $\style{font-family:'Times New Roman'}{A=\left[a_1,....,a_n\right]\;\mathrm{and}\;b=\sum_\limits{i=1}^na_i}$. The set of equations has 

Let u and v be two vectors in R2 whose Euclidean norms satisfy $\style{font-family:'Times New Roman'}{\parallel u\parallel=2\parallel v\parallel.}$ What is the value of  $\style{font-family:'Times New Roman'}\alpha$ such that  $\style{font-family:'Times New Roman'}{w=u+\alpha v}$ bisects the angle between u and v ?

Let A be $\style{font-family:'Times New Roman'}{n\times n}$ real valued  square symmetric matrix of rank 2 with $\style{font-family:'Times New Roman'}{\sum_\limits{i=1}^n\sum_\limits{j=1}^nA_{ij}^2=50.}$ Consider the following statements.
(I) One eigenvalue must be in [-5, 5]
(II) The eigenvalue with the largest magnitude must be strictly greater than 5
Which of the above statments about eigenvalue of A is/are necessarily CORRECT?

Let $ P=\begin{bmatrix}1&1&-1\\2&-3&4\\3&-2&3\end{bmatrix} $ and $ Q=\begin{bmatrix}-1&-2&-1\\6&12&6\\5&10&5\end{bmatrix} $ be two matrices.

Then the rank of P+Q is ________.

If the characteristics polynomial of a $\style{font-family:'Times New Roman'}{3\times3}$ matrix M over $\mathbb{R}$ (the set of real numbers) is $\style{font-family:'Times New Roman'}{\lambda^3-4\lambda^2+a\lambda+30,\;a\in\mathbb{R}}$ and one eigenvalue of M is 2, then the largest among the absolute values of the eigenvalues of M is _____________. 

Suppose that the eigenvalues of matrix A are 1,2,4. The determinant of $(A^{-1})^T$ is __________.

The larger of the two eigenvalues of the matrix 4521 is _____.

If the following system has non – trivial solution
px +qy + rz = 0
qx + ry + pz = 0
rx + py +qz = 0
then which one of the following Options is TRUE?

Consider the following system of equations:

3x + 2y = 1
4x + 7z = 1
x + y + z = 3
x – 2y + 7z = 0

The number of solutions for this system is __________________

The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4-by-4 symmetric positive definite matrix is ___________________.

If the matrix A is such that


then the determinant of A is equal to ______.

The product of the non-zero eigenvalues of the matrix


is ______.

Which one of the following statements is TRUE about every n × n matrix with only real eigenvalues?

If V1 and V2 are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dimension of V1V2 is ______.

Which one of the following does NOT equal 1xx21yy21zz2?

Let A be the 2 × 2 matrix with elements a11 = a12 = a21 = +1 and a22 = −1. Then the eigenvalues of the matrix A19 are

Consider the matrix as given below.


Which one of the following options provides the CORRECT values of the eigenvalues of the matrix?

Consider the following matrix


If the eigenvalues of A are 4 and 8, then

The following system of equations

x1+x2+2 x3 = 1
 x1+2 x2+3 x3 = 2
x1+4 x2+
α x3 = 4

has a unique solution. The only possible value(s) for α is/are

How many of the following matrices have an eigenvalue 1?

1000,0100,1-111  and -101-1

Let A be a 4 × 4 matrix with eigenvalues -5, -2, 1, 4. Which of the following is an eigenvalue of AIIA, where I is the 4 × 4 identity matrix?

Consider the set of (column) vectors defined by X=xR3|x1+x2+x3=0, where xT=x1,x2,x3T. Which of the following is TRUE?

Subjects of Engineering Mathematics