GATE Questions & Answers of Linear Algebra Electrical Engineering

Given a system of equations:

x+2y+2z=b1

5x+y+3z=b2

Which of the following is true regarding its solutions

A system matrix is given as follows

A=01-1-6-116-6-115

The absolute value of the ratio of the maximum eigenvalue to the minimum eigenvalue is _______

Which one of the following statements is true for all real symmetric matrices?

Two matrices A and B are given below:

A=pqrs; B=p2+q2pr+qspr+qsr2+s2

If the rank of matrix A is N, then the rank of matrix B is

The equation  2-21-1 x1x2=00 has

A matrix has eigenvalues –1 and –2. The corresponding eigenvectors are 1-1 and 1-2 respectively. The matrix is

Given that

A=-5-320 and I=1001, the value of A3 is

The matrix A=214-1  is decomposed into a product of a lower triangular matrix [L] and an upper triangular matrix [U]. The properly decomposed [L] and [U] matrices respectively are

An eigenvector of p=110022003 is

For the set of equations

x1 + 2x2 + x3 + 4x4 =2

3x1 + 6x2 + 3x2+ 12x4 = 6.

The following statement is true

The trace and determinant of a 2 × 2 matrix are known to be -2 and -35 respectively. Its eigen values are

The characteristic equation of a (3X3) matrix P is defined as

αλ=|λI-P|=λ3+λ2+1=0

If I denotes identity matrix, then the inverse of  matrix P will be

If the rank of a (5X6) matrix Q is 4, then which one of the following statement is correct ?

A is a m x n full rank matrix with m>n and I is identity matrix. Let matrix A+=(ATA)-1AT,Then, which one of the following statement is FALSE ?

Let P be a 2 X 2 real orthogonal matrix and x is a real vector x1,x2τ with length $\style{font-size:14px}{\parallel\overrightarrow x\parallel=\left(x_1^2+x_2^2\right)^\frac12.}$ Then, which one of the following statements is correct ?

x = [x1 x2 ... xn]T is an n-tuple nonzero vector. The n×n matrix  V=xx T

Let x and y be two vectors in a 3 dimensional space and <x, y> denote their dot product. Then the determinant

det<x,x><x,y><y,x><y,y>

The linear operation L(x) is defined by the cross product L(x) = bXx, where b=[0 1 0]T and x=[x1 x2 x3]T are three dimensional vectors. The 3×3 matrix M of this operations satisfies Lx=Mx1x2x3

Then the eigenvalues of M are

Cayley-Hamilton Theorem states that a square matrix satisfies its own characteristic equation. Consider a matrix A=-32-10

A satisfies the relation

Cayley-Hamilton Theorem states that a square matrix satisfies its own characteristic equation. Consider a matrix A=-32-10

A9 equals