# GATE Questions & Answers of Linear Algebra Civil Engineering

#### Linear Algebra 28 Question(s)

The matrix $\begin{pmatrix}2&-4\\4&-2\end{pmatrix}$ has

The matrix P is the inverse of a matrix Q. If I denotes the identitiy matrix, which one of the following option is correct?

The number of parameters in the univariate exponential and Gaussian distributions, respectively, are

Consider the matrix $\style{font-family:'Times New Roman'}{\begin{bmatrix}5&-1\\4&1\end{bmatrix}}$ Which one of the following statements is TRUE for the eigenvalues and eigenvectors of this matrix?

Consider following simultaneous equations (with $\begin{array}{l}c_1\\\end{array}$ and $\begin{array}{l}c_2\\\end{array}$ being constants):

$\begin{array}{l}3\;x_1+2\;x_2=c_1\\4\;x_1+x_2=c_2\end{array}$

The characteristic equation for these simultaneous equation is

If $\mathbf A=\begin{bmatrix}1&5\\6&2\end{bmatrix}$ and $\mathbf B=\begin{bmatrix}3&7\\8&4\end{bmatrix},\boldsymbol A\boldsymbol B^T$ is equal to

If the entries in each column of a square matrix $Μ$ add up to 1, then an eigenvalue of $Μ$ is

Consider the following linear system.

$x+2y-3z=a$

$2x+3y+3z=b$

$5x+9y-6z=c$

This system is consistent if $a, b$ and $c$ satisfy the equation

For what value of p the following set of equation will have no solution ?

$\begin{array}{l}2\mathrm x+3\mathrm y=5\\3\mathrm x+\mathrm{py}=10\end{array}$

The smallest and largest Eigen values of the following matrix are:

$\left[\begin{array}{ccc}3& -2& 2\\ 4& -4& 6\\ 2& -3& 5\end{array}\right]$

Let $A=\lbrack a_{ij}\rbrack,1\leq i,j<n$ with $n\geq3$ and $a_{ij}=i.j.$ The rank of $A$ is :

The two Eigen Values of the matrix $\left[\begin{array}{cc}2& 1\\ 1& p\end{array}\right]$ have a ratio of 3:1 for p = 2. What is another value of p for which the Eigen values have the same ratio of 3:1?

Given the matrices $J=\left[\begin{array}{ccc}3& 2& 1\\ 2& 4& 2\\ 1& 2& 6\end{array}\right]$ and $K=\left[\begin{array}{c}1\\ 2\\ -1\end{array}\right]$,the product KT JK is _________.

The sum of Eigen values of the matrix, [M] is

The determinant of matrix $\left[\begin{array}{cccc}0& 1& 2& 3\\ 1& 0& 3& 0\\ 2& 3& 0& 1\\ 3& 0& 1& 2\end{array}\right]$ is ____________

The rank of the matrix $\left[\begin{array}{cccc}6& 0& 4& 4\\ -2& 14& 8& 18\\ 14& -14& 0& -10\end{array}\right]$ is _______________

What is the minimum number of multiplications involved in computing the matrix product PQR? Matrix P has 4 rows and 2 columns, matrix Q has 2 rows and 4 columns, and matrix R has 4 rows and 1 column. __________

The eigen values of matrix $\left[\begin{array}{cc}9& 5\\ 5& 8\end{array}\right]$ are

[A] is a square matrix which is neither symmetric nor skew-symmetric and [A]T is its transpose.The sum and differene of these matrices are defined as [s]=[A]+[A]T and [D]=[A]-[A]T,respectively.Which of the following statements is true?

The inverse of the matrix  $\left[\begin{array}{cc}3+2i& i\\ -i& 3-2i\end{array}\right]$ is

A square matrix B is skew-symmetric if

The product of matrices is

The value of $\int\limits_0^3\int\limits_0^x\left(6-x-y\right)dx\;dy$ is

The Eigen values of the matrix $\left[P\right]=\left[\begin{array}{cc}4& 5\\ 2& -5\end{array}\right]$ are

The following simultaneous equations

x+y+z=3
x+2y+3z=4
x+4y+kz=6

will NOT have a unique solution for k equal to

The minimum and the maximum eigen velue of the matrix $\left[\begin{array}{ccc}1& 1& 3\\ 1& 5& 1\\ 3& 1& 1\end{array}\right]$ are -2 and 6, respectively. What is the other eigen value?

For what values of $\mathrm{\alpha }$ and $\mathrm{\beta }$ the following simultaneous equations have an infinite number of solutions?
x + y + z = 5;     x + 3y + 3z = 9;     x + 2y + $\mathrm{\alpha }$z = $\mathrm{\beta }$
The inverse of the 2 × 2 matrix $\left[\begin{array}{cc}1& 2\\ 5& 7\end{array}\right]$ is