# GATE Questions & Answers of Linear Algebra Electrical Engineering

#### Linear Algebra 20 Question(s)

Given a system of equations:

$x+2y+2z={b}_{1}$

$5x+y+3z={b}_{2}$

Which of the following is true regarding its solutions

A system matrix is given as follows

$A=\left[\begin{array}{ccc}0& 1& -1\\ -6& -11& 6\\ -6& -11& 5\end{array}\right]$

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:

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

The equation  has

A matrix has eigenvalues –1 and –2. The corresponding eigenvectors are $\left[\begin{array}{c}1\\ -1\end{array}\right]$ and $\left[\begin{array}{c}1\\ -2\end{array}\right]$ respectively. The matrix is

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 matrix $\left[A\right]=\left[\begin{array}{cc}2& 1\\ 4& -1\end{array}\right]$  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 $\mathrm{p}=\left(\begin{array}{ccc}1& 1& 0\\ 0& 2& 2\\ 0& 0& 3\end{array}\right)$ 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

$\alpha \left(\lambda \right)=|\lambda \mathbf{I}-\mathbf{P}|={\lambda }^{3}+{\lambda }^{2}+1=0$

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 $\stackrel{\mathbf{\to }}{\mathbf{x}}$ is a real vector ${\left[{x}_{1},{x}_{2}\right]}^{\tau }$ 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

$\mathrm{det}\left[\begin{array}{cc}& \\ & \end{array}\right]$

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 $\mathrm{L}\left(\mathrm{x}\right)=\mathrm{M}\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right]$

Then the eigenvalues of M are

Cayley-Hamilton Theorem states that a square matrix satisfies its own characteristic equation. Consider a matrix $A=\left[\begin{array}{cc}-3& 2\\ -1& 0\end{array}\right]$

A satisfies the relation

Cayley-Hamilton Theorem states that a square matrix satisfies its own characteristic equation. Consider a matrix $A=\left[\begin{array}{cc}-3& 2\\ -1& 0\end{array}\right]$