# GATE Questions & Answers of Linear Algebra Electronics and Communication Engg

#### Linear Algebra 28 Question(s)

Consider the 5 x 5 matrix

$A=\left[\begin{array}{ccccc}1& 2& 3& 4& 5\\ 5& 1& 2& 3& 4\\ 4& 5& 1& 2& 3\\ 3& 4& 5& 1& 2\\ 2& 3& 4& 5& 1\end{array}\right]$

It is given that A has only one real eigenvalue. Then the real eigenvalue of A is

The rank of the matrix $M=\left[\begin{array}{ccc}5& 10& 10\\ 1& 0& 2\\ 3& 6& 6\end{array}\right]$ is

The rank of the matrix $\left[\begin{array}{ccccc}1& -1& 0& 0& 0\\ 0& 0& 1& -1& 0\\ 0& 1& -1& 0& 0\\ -1& 0& 0& 0& 1\\ 0& 0& 0& 1& -1\end{array}\right]$ is_________.

L et M4 = I, (where I denotes the identity matrix) and M ≠ I, M2 ≠ I and M3 ≠ I. Then, for any natural number k, M−1 equals:

A sequence $x\left[n\right]$ is specified as

for $n\ge 2.$

The initial conditions are , and $x\left[n\right]=0$ for $n<0$. The value of $x\left[12\right]$ is ______

The value of $x$ for which the matrix $A=\left[\begin{array}{ccc}3& 2& 4\\ 9& 7& 13\\ -6& -4& -9+x\end{array}\right]$

has zero as an eigenvalue is ________

The matrix $A=\begin{bmatrix}a&0&3&7\\2&5&1&3\\0&0&2&4\\0&0&0&b\end{bmatrix}$ has det $(A) = 100$ and trace $(A) = 14.$

The value of $\left|a-b\right|$ is ________

Consider a 2 × 2 square matrix

where x is unknown. If the eigenvalues of the matrix A are $\left(\sigma +j\omega \right)$ and $\left(\sigma -j\omega \right)$ , then x is equal to

Consider a system of linear equations :

x2y +3z = –1
x3y + 4z = 1 and
2x +4y3z = k.

The value of k for which the system has infinitely many solutions is _______.

The value of $p$ such that the vector $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$ is an eigenvector of the matrix $\begin{bmatrix}4&1&2\\p&2&1\\14&-4&10\end{bmatrix}$ is_______.

The value of $x$ for which all the eigen-values of the matrix given below are real is

$\left[\begin{array}{ccc}10& 5+j& 4\\ x& 20& 2\\ 4& 2& -10\end{array}\right]$

For $A=\left[\begin{array}{cc}1& \mathrm{tan}x\\ -\mathrm{tan}x& 1\end{array}\right]$, the determinant of AT A-1 is

For matrices of same dimension M, N and scalar c, which one of these properties DOES NOT ALWAYS hold?

A real (4 × 4) matrix A satisfies the equation A2=I, where is the (4 × 4) identity matrix. The positive eigen value of A is _____.

Consider the matrix

${J}_{6}=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\end{array}\right]$

which is obtained by reversing the order of the columns of the identity matrix I6.

Let $P={I}_{6}+\alpha {J}_{6,}$ where $\alpha$ is a non-negative real number. The value of $\alpha$ for which det (P) = 0 is _____

The determinant of matrix A is 5 and the determinant of matrix B is 40. The determinant of matrix AB  is ________.

The system of linear equations

$\left(\begin{array}{ccc}2& 1& 3\\ 3& 0& 1\\ 1& 2& 5\end{array}\right)\left(\begin{array}{c}a\\ b\\ c\end{array}\right)=\left(\begin{array}{c}5\\ -4\\ 14\end{array}\right)$ has

The maximum value of the determinant among all 2×2 real symmetric matrices with trace 14 is ________.

Which one of the following statements is NOT true for a square matrix A?

The minimum eigenvalue of the following matrix is $\left[\begin{array}{ccc}3& 5& 2\\ 5& 12& 7\\ 2& 7& 5\end{array}\right]$

Let A be an m x n matrix and B an n x m matrix. It is given that determinant (Im + AB) = determinant (In +BA) , where Ik is the k x k identity matrix. Using the above property, the determinant of the matrix given below is

$\left[\begin{array}{cccc}2& 1& 1& 1\\ 1& 2& 1& 1\\ 1& 1& 2& 1\\ 1& 1& 1& 2\end{array}\right]$

Given that

$\mathbit{A}=\left[\begin{array}{cc}-5& -3\\ 2& 0\end{array}\right]$ and $\mathbit{I}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, the value of A3 is

The system of equations
x +y+ z= 6
x+ 4y+ 6z = 20
x+ 4y+ λz = μ

has NO solution for values of λ and μ given by

The eigen values of a skew-symmetric matrix are

The eigen values of the following matrix are $\left[\begin{array}{ccc}-1& 3& 5\\ -3& -1& 6\\ 0& 0& 3\end{array}\right]$

All the four entries of the 2×2 matrix $p=\left[\begin{array}{cc}{p}_{11}& {p}_{12}\\ {p}_{21}& {p}_{22}\end{array}\right]$ are nonzero, and one of its eigenvalues is zero. Which of the following statements is true?

The system of linear equations

4x + 2y = 7
2x + y = 6

has

Consider the matrix $\mathbf{P}=\left[\begin{array}{cc}0& 1\\ -2& -3\end{array}\right]$. The value of ep is