# 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 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.