GATE Questions & Answers of Linear Algebra Civil Engineering

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

•  (A) real eigenvalues and eigenvectors
•  (B) real eigenvalues but complex eigenvectors
•  (C) complex eigenvalues but real eigenvectors
•  (D) complex eigenvalues and eigenvectors

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

•  (A)  $\mathrm{PQ}\;=\mathrm I $ but $ \mathrm{QP}\;\neq\;\mathrm I $ 
•  (B)  $ \mathrm{QP}\;=\mathrm I $ but $ \mathrm{PQ}\;\neq\;\mathrm I $
•  (C)  $ \mathrm{PQ}\;=\mathrm I $ and $ \mathrm{QP}\;=\;\mathrm I $
•  (D)  $ \mathrm{PQ}\;-\;\mathrm{QP}\;=\;\mathrm I $

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

•  (A)  2 and 2
•  (B)  1 and 2
•  (C)  2 and 1
•  (D)  1 and 1

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?

•  (A)  Eigenvalue 3 has multiplicity of 2, and only one independent eigenvector exists.
•  (B)  Eigenvalue 3 has multiplicity of 2, and two  independent eigenvectors exist.
•  (C)  Eigenvalue 3 has multiplicity of 2, and no independent eigenvector exists.
•  (D)  Eigenvalues are  3 and -3, and two independent eigenvectors exist.

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

•  (A)  $\style{font-family:'Times New Roman'}{\lambda^2-4\lambda-5=0}$
•  (B)  $\style{font-family:'Times New Roman'}{\lambda^2-4\lambda+5=0}$
•  (C)  $\style{font-family:'Times New Roman'}{\lambda^2+4\lambda-5=0}$
•  (D)  $\style{font-family:'Times New Roman'}{\lambda^2+4\lambda+5=0}$

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

•  (A)  $\style{font-family:'Times New Roman'}{\begin{bmatrix}38&28\\32&56\end{bmatrix}}$
•  (B)  $\style{font-family:'Times New Roman'}{\begin{bmatrix}3&40\\42&8\end{bmatrix}}$
•  (C)  $\style{font-family:'Times New Roman'}{\begin{bmatrix}43&27\\34&50\end{bmatrix}}$
•  (D)  $\style{font-family:'Times New Roman'}{\begin{bmatrix}38&32\\28&56\end{bmatrix}}$

If the entries in each column of a square matrix ${\rm M}$ add up to 1, then an eigenvalue of ${\rm M}$ is

•  (A)  4
•  (B)  3
•  (C)  2
•  (D)  1

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

•  (A)  $ 7a-b-c=0 $
•  (B)  $ 3a+b-c=0 $
•  (C)  $ 3a-b+c=0 $
•  (D)  $ 7a-b+c=0 $

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:

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

•  (A)  1.5 and 2.5
•  (B)  0.5 and 2.5
•  (C)  1.0 and 3.0
•  (D)  1.0 and 2.0

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

•  (A)  0
•  (B)  1
•  (C)  $ n-1 $
•  (D)  $ n $

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?

•  (A)  -2
•  (B)  1
•  (C)  7/3
•  (D)  14/3

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 K^T JK is _________.

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

$where \left[M\right]=\left[\begin{array}{ccc}215& 650& 795\\ 655& 150& 835\\ 485& 355& 550\end{array}\right]$

•  (A) 915
•  (B) 1355
•  (C) 1640
•  (D) 2180

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) -2.42 and 6.86
•  (B) 3.48 and 13.53
•  (C) 4.70 and 6.86
•  (D) 6.86 and 9.50

[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?

•  (A) Both [S] and [D] are symmetric
•  (B) Both[S] and [D] are skew-symmetric
•  (C) [S] is skew-symmetric and [D] is symmetric
•  (D) [S] is symmetric and [D] is skew-symmetric

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

•  (A) $\frac{1}{12}\left[\begin{array}{cc}3+2i& -i\\ i& 3-2i\end{array}\right]$
•  (B) $\frac{1}{12}\left[\begin{array}{cc}3-2i& -i\\ i& 3+2i\end{array}\right]$
•  (C) $\frac{1}{14}\left[\begin{array}{cc}3+2i& -i\\ i& 3-2i\end{array}\right]$
•  (D) $\frac{1}{14}\left[\begin{array}{cc}3-2i& -i\\ i& 3+2i\end{array}\right]$

A square matrix B is skew-symmetric if

•  (A) B^T = -B
•  (B) B^T = B
•  (C) B^-1 = B
•  (D) B^-1 = B^T

The product of matrices ${\left(PQ\right)}^{-1} P$ is

•  (A) $P^{-1}$;
•  (B) $Q^{-1}$;
•  (C) $P^{-1 }{Q}^{-1} P$;
•  (D) $PQ P^{-1}$;

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

•  (A) 13.5
•  (B) 27.0
•  (C) 40.5
•  (D) 54.0

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

•  (A) -7 and 8
•  (B) -6 and 5
•  (C) 3 and 4
•  (D) 1 and 2

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

•  (A) 0
•  (B) 5
•  (C) 6
•  (D) 7

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?

•  (A) 5
•  (B) 3
•  (C) 1
•  (D) -1

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 = $\mathrm{\beta }$

•  (A) 2, 7
•  (B) 3, 8
•  (C) 8, 3
•  (D) 7, 2

The inverse of the 2 × 2 matrix $\left[\begin{array}{cc}1& 2\\ 5& 7\end{array}\right]$ is

•  (A) $\frac{1}{3}\left[\begin{array}{cc}-7& 2\\ 5& -1\end{array}\right]$
•  (B) $\frac{1}{3}\left[\begin{array}{cc}7& 2\\ 5& 1\end{array}\right]$
•  (C) $\frac{1}{3}\left[\begin{array}{cc}7& -2\\ -5& 1\end{array}\right]$
•  (D) $\frac{1}{3}\left[\begin{array}{cc}-7& -2\\ -5& -1\end{array}\right]$