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

#### Linear Algebra 27 Question(s)

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 $\left[\begin{array}{cc}4& 5\\ 2& 1\end{array}\right]$ 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

$A=\left[\begin{array}{c}2\\ -4\\ 7\end{array}\right]\left[\begin{array}{ccc}1& 9& 5\end{array}\right]$

then the determinant of A is equal to ______.

The product of the non-zero eigenvalues of the matrix

$\left[\begin{array}{ccccc}1& 0& 0& 0& 1\\ 0& 1& 1& 1& 0\\ 0& 1& 1& 1& 0\\ 0& 1& 1& 1& 0\\ 1& 0& 0& 0& 1\end{array}\right]$

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 $\left|\begin{array}{ccc}1& x& {x}^{2}\\ 1& y& {y}^{2}\\ 1& z& {z}^{2}\end{array}\right|$?

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.

$\left[\begin{array}{ccc}1& 2& 3\\ 0& 4& 7\\ 0& 0& 3\end{array}\right]$

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

Consider the following matrix

$A=\left[\begin{array}{cc}2& 3\\ x& y\end{array}\right]$

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+
$\alpha$ x3 = 4

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

How many of the following matrices have an eigenvalue 1?

Let A be a 4 × 4 matrix with eigenvalues -5, -2, 1, 4. Which of the following is an eigenvalue of $\left[\begin{array}{cc}A& I\\ I& A\end{array}\right]$, where I is the 4 × 4 identity matrix?