GATE Questions & Answers of Linear Algebra Electronics and Communication Engg

Consider the 5 x 5 matrix

A=1234551234451233451223451

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

The rank of the matrix M=51010102366 is

The rank of the matrix 1-1000001-1001-100-100010001-1 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 xn is specified as

xnxn-1=1110n10, for n2.

The initial conditions are x0=1, x1=1, and xn=0 for $n<0$. The value of x12 is ______

The value of $x$ for which the matrix A=3249713-6-4-9+x
 
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 a-b is ________

Consider a 2 × 2 square matrix

        A=σxωσ

where x is unknown. If the eigenvalues of the matrix A are σ+jω and σ-jω , 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 123 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

105+j4x20242-10

For A=1tanx-tanx1, 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

J6=000001000010000100001000010000100000

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

Let P=I6+αJ6, where α is a non-negative real number. The value of α 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

213301125abc=5-414 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 3525127275

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

2111121111211112  

Given that

A=-5-320 and I=1001, 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 -135-3-16003

All the four entries of the 2×2 matrix p=p11p12p21p22 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 P=01-2-3. The value of ep is