Linear Algebra

Question 1
The number of purely real elements in a lower triangular representation of the given 3x3 matrix, obtained through the given decomposition is
\begin{bmatrix} 2 &3 &3 \\ 3& 2 & 1\\ 3& 1 & 7 \end{bmatrix}= \begin{bmatrix} a_{11} &0 &0 \\ a_{12}& a_{22} & 0\\ a_{13}& a_{23} & a_{33} \end{bmatrix}\begin{bmatrix} a_{11} &0 &0 \\ a_{12}& a_{22} & 0\\ a_{13}& a_{23} & a_{33} \end{bmatrix}^T
A
5
B
6
C
8
D
9
GATE EE 2020   Engineering Mathematics
Question 1 Explanation: 
As per GATE official answer key MTA (Marks to ALL)
\begin{aligned} \begin{bmatrix} 2 &3 &3 \\ 3 &2 &1 \\ 3 &1 &7 \end{bmatrix}&=\begin{bmatrix} I_{11} & 0 &0 \\ I_{21}&l_{22} & 0\\ l_{31} &l_{32} &l_{33} \end{bmatrix}\begin{bmatrix} u_{11} &u_{12} &u_{13} \\ 0 &u_{22} &u_{23} \\ 0 & 0 &u_{33} \end{bmatrix} \\ \text{consider, }u_{11}&=u_{22}=u_{33}=1 \\ \begin{bmatrix} 2 &3 &3 \\ 3 &2 &1 \\ 3 &1 &7 \end{bmatrix}&=\begin{bmatrix} l_{11} & 0 &0 \\ l_{21}&l_{22} & 0\\ l_{31} &l_{32} &l_{33} \end{bmatrix}\begin{bmatrix} 1 &u_{12} &u_{13} \\ 0 &1 &u_{23} \\ 0 & 0 &1 \end{bmatrix} \\ \begin{bmatrix} 2 &3 &3 \\ 3 &2 &1 \\ 3 &1 &7 \end{bmatrix}&=\begin{bmatrix} l_{11} & l_{11}u_{12} &l_{11}u_{13} \\ l_{21}& l_{21}u_{12}+l_{22} & l_{21}u_{13}+ l_{22}u_{23}\\ l_{31} &l_{31}u_{12}+l_{32} & l_{31}u_{13}+ l_{32}u_{23}+l_{33}\end{bmatrix} \\ l_{11}&=2 \; \; l_{11}u_{12}=3 \; \; l_{11}u_{13}=3 \; \; \\ l_{21}&=3 \; \; 2u_{12}=3 \; \; 2u_{13}=3 \\ l_{31}&=3\; \; u_{12}=\frac{3}{2} \; \; u_{13}=\frac{3}{2} \\ l_{21}u_{12}+l_{22}&=2\; \; l_{21}u_{13}+l_{22}u_{23}=1 \\ (3)\left ( \frac{3}{2} \right )+l_{22}&=2 \; \; (3)\left ( \frac{3}{2} \right )+\left ( -\frac{5}{2} \right )u_{23}=1 \\ l_{22}&=-\frac{5}{2} \; \; u_{23}=-\frac{7}{5} \\ l_{31}u_{12}+l_{32}&=1 \; \; l_{31}u_{13}+l_{32}u_{23}+l_{33}=7\\ (3)(\frac{3}{2})+l_{32}&=1 \; \; (3)\left ( \frac{3}{2} \right )+\left ( -\frac{7}{2} \right )\left ( \frac{7}{5} \right )+l_{33}=7 \\ l_{32}&=-\frac{7}{2} \; \; l_{33}=-\frac{74}{10}\\ L&=\begin{bmatrix} 2 &0 &0 \\ 3 &-5/2 &0 \\ 3 &-7/2 & 74/10 \end{bmatrix} \end{aligned}
The number of purely real elements of lower triangular matrix are 9.
Question 2
Consider a 2x2 matrix M=[v_1\;\; v_2], where, v_1\;and \; v_2 are the column vectors. Suppose M^{-1}=\begin{bmatrix} u_1^T\\ u_2^T \end{bmatrix}, where u_1^T \; and \; u_2^T are the row vectors. Consider the following statements:

Statement 1: u_1^T v_1=1\; and \; u_2^Tv_2=1
Statement 2: u_1^T v_2=0\; and \; u_2^Tv_1=0

Which of thefollowing options is correct?
A
Statement 1 is true and statement 2 is false
B
Statement 2 is true and statement 1 is false
C
Both the statements are true
D
Both the statements are false
GATE EE 2019   Engineering Mathematics
Question 2 Explanation: 
\begin{aligned} M^{-1}M &=I \\ \Rightarrow \; \begin{bmatrix} u_1^T\\ u_2^T \end{bmatrix}\begin{bmatrix} v_1 & v_2 \end{bmatrix} &= \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\\ \begin{bmatrix} u_1^T v_1 & u_1^T v_2\\ u_2^T v_1 & u_2^T v_2 \end{bmatrix}&=\begin{bmatrix} 1 &0 \\ 0 & 1 \end{bmatrix} \\ u_1^T v_1=1; &u_1^T v_2 =0\\ u_2^T v_1=1; &u_2^T v_2 =1\\ \end{aligned}
Statement (1) and (2) are both correct. Option (C) is correct.
Question 3
The rank of the matrix, M=\begin{bmatrix} 0 &1 &1 \\ 1& 0& 1\\ 1& 1& 0 \end{bmatrix}, is ______
A
1
B
2
C
3
D
4
GATE EE 2019   Engineering Mathematics
Question 3 Explanation: 
\begin{aligned} M &=\begin{bmatrix} 0 & 1 & 1\\ 1& 0 & 1\\ 1& 1 & 0 \end{bmatrix} \\ &R_1\leftrightarrow R_2 \\ &=\begin{bmatrix} 1 & 0 & 1\\ 0& 1 & 1\\ 1& 1 & 0 \end{bmatrix} \\ &R_3\rightarrow R_3 -R_1 \\ &= \begin{bmatrix} 1 & 0 & 1\\ 0& 1 & 1\\ 0& 1 & -1 \end{bmatrix}\\ &R_3\rightarrow R_3-R_2 \\ &= \begin{bmatrix} 1 & 0 & 1\\ 0& 1 & 1\\ 0& 0 & -2 \end{bmatrix} \end{aligned}
Which is in echelon form
\therefore \;\; \rho (A)=3
Question 4
M is a 2x2 matrix with eigenvalues 4 and 9. The eigenvalues of M^2 are
A
4 and 9
B
2 and 3
C
-2 and -3
D
16 and 81
GATE EE 2019   Engineering Mathematics
Question 4 Explanation: 
M is 2 x 2 matrix with eigen values 4 and 9. The eigen values of M^2 are 16 and 81.
Question 5
Let A=\begin{bmatrix} 1 & 0&-1 \\ -1&2 & 0\\ 0& 0 & -2 \end{bmatrix} and B=A^{3}-A^{2}-4A+5I, where I is the 3x3 identity matrix. The determinant of B is ________ (up to 1 decimal place).
A
0.5
B
1.0
C
2.8
D
2.1
GATE EE 2018   Engineering Mathematics
Question 5 Explanation: 
\begin{aligned} A=\begin{bmatrix} 1 & 0 & -1\\ -1 &2 &0 \\ 0 & 0 & -2 \end{bmatrix}\\ |A-\lambda I|=0\\ \begin{bmatrix} 1-\lambda & 0& -1\\ -1& 2-\lambda & 0\\ 0 & 0 &-2-\lambda \end{bmatrix}=0\\ (1-\lambda )((2-\lambda )(-2-\lambda ))-1(0-0)=0\\ \lambda =1,2,-2\\ \text{Eigen value of A are 1,2,-2}\\ \text{Eigen value of } A^3 \text{ are 1,8,-8}\\ \text{Eigen value of } A^2 \text{ are 1,4,4}\\ \text{Eigen value of } 4A \text{ are 4,8,-8}\\ \text{Eigen value of } 5I \text{ are 5,5,5}\\ A^3-A^2-4A+5I \text{ are 1,1,1}\\ \therefore \;\; |B|=(1)(1)(1)=1 \end{aligned}
Question 6
Consider a non-singular 2x2 square matrix A. If trace(A)=4 and trace(A^{2})=5, the determinant of the matrix A is _________(up to 1 decimal place).
A
2.5
B
3.5
C
1.2
D
5.5
GATE EE 2018   Engineering Mathematics
Question 6 Explanation: 
A is 2 x 2 matrix
\begin{aligned} Tr\; A&=4\\ \lambda _1+\lambda _2&=4\\ tr(A^2)&=5\\ \lambda _1^2+\lambda _2^2&=5\\ (\lambda _1+\lambda _2)^2&=\lambda _1^2+\lambda _2^2+2\lambda _1 \lambda _2\\ 2\lambda _1 \lambda _2+5&=16\\ 2\lambda _1 \lambda _2&=11\\ \lambda _1 \lambda _2&=\frac{11}{2}\\ |A|&=\frac{11}{2}=5.5 \end{aligned}
Question 7
The eigen values of the matrix given below are \begin{bmatrix} 0 & 1&0 \\ 0& 0 & 1\\ 0 &-3 & -4 \end{bmatrix}
A
(0,-1,-3)
B
(0,-2,-3)
C
(0,2,3)
D
(0,1,3)
GATE EE 2017-SET-2   Engineering Mathematics
Question 7 Explanation: 
The characteristics equation is |A-\lambda |i=0.
\begin{aligned} \begin{bmatrix} 0-\lambda & 1 & 0\\ 0 & 0-\lambda & 1\\ 0& -3 &-4-\lambda \end{bmatrix}&=0 \\ -\lambda (4\lambda +\lambda ^2+3)-1(0-0)&=0 \\ -\lambda (\lambda ^2+4\lambda +3)&=0 \\ \lambda =0,(\lambda +1)(\lambda +3)&=0 \\ \lambda &=-1,-3 \\ \lambda &=(0,-1,-3) \end{aligned}
Question 8
The matrix A=\begin{bmatrix} \frac{3}{2} & 0 & \frac{1}{2}\\ 0 & -1 & 0\\ \frac{1}{2}& 0 & \frac{3}{2} \end{bmatrix} has three distinct eigenvalues and one of its eigenvectors is \begin{bmatrix} 1\\ 0\\ 1 \end{bmatrix}
Which one of the following can be another eigenvector of A?
A
\begin{bmatrix} 0\\ 0\\ -1 \end{bmatrix}
B
\begin{bmatrix} -1\\ 0\\ 0 \end{bmatrix}
C
\begin{bmatrix} 1\\ 0\\ -1 \end{bmatrix}
D
\begin{bmatrix} 1\\ -1\\ 1 \end{bmatrix}
GATE EE 2017-SET-1   Engineering Mathematics
Question 8 Explanation: 
The given matrix is symmetric and all its eigen values are distinct. Hence all its eigen vectors are orthogonal one of the eigen vector is
x_1=\begin{bmatrix} 1\\ 0\\ 1 \end{bmatrix}
The corresponding orthogonal vector in the given option is C i.e. x_2=\begin{bmatrix} 1\\ 0\\ -1 \end{bmatrix}
{x_1}^T\cdot x_2=\begin{bmatrix} 1 & 0 & 1 \end{bmatrix}\begin{bmatrix} 1\\ 0\\ -1 \end{bmatrix}=1+0-1=0
Question 9
Let P=\begin{bmatrix} 3 & 1\\ 1&3 \end{bmatrix}. Consider the set S of all vectors \binom{x}{y} such that a^{2}+b^{2}=1 where \binom{a}{b} =P\binom{x}{y}. Then S is
A
a circle of radius \sqrt{10}
B
a circle of radius \frac{1}{\sqrt{10}}
C
an ellipse with major axis along \binom{1}{1}
D
an ellipse with minor axis along \binom{1}{1}
GATE EE 2016-SET-2   Engineering Mathematics
Question 9 Explanation: 
\begin{aligned} \begin{bmatrix} 3 & 1\\ 1 & 3 \end{bmatrix}\begin{bmatrix} x\\ y \end{bmatrix}&=\begin{bmatrix} a\\ b \end{bmatrix} \\ 3x+y &=a \\ x+3y&=b \\ a^2+b^2&=1 \\ \Rightarrow \;\; 10x^2+10y^2+12xy &= 1 \end{aligned}
Ellipse with major axis along \begin{bmatrix} 1\\ 1 \end{bmatrix}.
Question 10
A 3 x 3 matrix P is such that, P^{3} = P. Then the eigenvalues of P are
A
1, 1, -1
B
1, 0.5 + j0.866, 0.5 - j0.866
C
1, -0.5 + j0.866, -0.5 - j0.866
D
0, 1, -1
GATE EE 2016-SET-2   Engineering Mathematics
Question 10 Explanation: 
By Calyey Hamilton theorem,
\lambda ^3=\lambda
\lambda =0, 1, -1
There are 10 questions to complete.
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