# Linear Algebra

 Question 1
Which of the following is/are the eigenvector(s) for the matrix given below?
$\begin{pmatrix} -9 &-6 &-2 &-4 \\ -8& -6 & -3 & -1 \\ 20 & 15 & 8 & 5 \\ 32& 21& 7&12 \end{pmatrix}$
MSQ
 A $\begin{pmatrix} -1\\ 1\\ 0\\ 1 \end{pmatrix}$ B $\begin{pmatrix} 1\\ 0\\ -1\\ 0 \end{pmatrix}$ C $\begin{pmatrix} -1\\ 0\\ 0\\ 0 \end{pmatrix}$ D $\begin{pmatrix} 0\\ 1\\ -3\\ 0 \end{pmatrix}$
GATE CSE 2022   Engineering Mathematics
Question 1 Explanation:
 Question 2
Consider solving the following system of simultaneous equations using LU decomposition.
\begin{aligned} x_1+x_2-2x_3&=4 \\ x_1+3x_2-x_3&=7 \\ 2x_1+x_2-5x_3&=7 \end{aligned}
where L and U are denoted as
$L= \begin{bmatrix} L_{11} & 0 & 0 \\ L_{21}& L_{22} & 0 \\ L_{31} & L_{32} & L_{33} \end{bmatrix}, U= \begin{bmatrix} U_{11} & U_{12} & U_{13} \\ 0& U_{22} & U_{23} \\ 0 & 0 & U_{33} \end{bmatrix}$
Which one of the following is the correct combination of values for $L32, U33,$ and $x_1$?
 A $L_{32}=2,U_{33}=-\frac{1}{2},x_1=-1$ B $L_{32}=2,U_{33}=2,x_1=-1$ C $L_{32}=-\frac{1}{2},U_{33}=2,x_1=0$ D $L_{32}=-\frac{1}{2},U_{33}=-\frac{1}{2},x_1=0$
GATE CSE 2022   Engineering Mathematics
Question 2 Explanation:
 Question 3
Consider the following two statements with respect to the matrices $A_{m \times n},B_{n \times m},C_{n \times n} \text{ and }D_{n \times n},$

Statement 1: $tr(AB) = tr(BA)$
Statement 2: $tr(CD) = tr(DC)$

where$tr()$ represents the trace of a matrix. Which one of the following holds?
 A Statement 1 is correct and Statement 2 is wrong. B Statement 1 is wrong and Statement 2 is correct. C Both Statement 1 and Statement 2 are correct. D Both Statement 1 and Statement 2 are wrong.
GATE CSE 2022   Engineering Mathematics
Question 3 Explanation:
 Question 4
Suppose that P is a 4x5 matrix such that every solution of the equation Px=0 is a scalar multiple of $\begin{bmatrix} 2 & 5 & 4 &3 & 1 \end{bmatrix}^T$. The rank of P is _______
 A 1 B 2 C 3 D 4
GATE CSE 2021 SET-2   Engineering Mathematics
Question 4 Explanation:
 Question 5
Consider the following matrix.
$\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\ 1& 1 & 0 & 1 \\1 & 1 & 1 & 0 \end{pmatrix}$
The largest eigenvalue of the above matrix is __________.
 A 1 B 3 C 4 D 6
GATE CSE 2021 SET-1   Engineering Mathematics
Question 5 Explanation:
 Question 6
If $x+2 y=30$,then $\left(\frac{2 y}{5}+\frac{x}{3}\right)+\left(\frac{x}{5}+\frac{2 y}{3}\right)$ will be equal to
 A 8 B 16 C 18 D 20
ISRO CSE 2020   Engineering Mathematics
Question 6 Explanation:
 Question 7
Let A and B be two nxn matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.

I. rank(AB)=rank (A)rank (B)
II. det(AB)=det(A)det(B)
III. rank(A+B) $\leq$ rank (A) + rank (B)
IV. det(A+B) $\leq$ det(A) + det(B)

Which of the above statements are TRUE?
 A I and II only B I and IV only C II and III only D III and IV only
GATE CSE 2020   Engineering Mathematics
Question 7 Explanation:
 Question 8
Consider the following matrix:
$\begin{bmatrix} 1 & 2 & 4 & 8\\ 1& 3 & 9 &27 \\ 1 & 4 & 16 &64 \\ 1 & 5 & 25 &125 \end{bmatrix}$
The absolute value of the product of Eigenvalues of R is _________ .
 A 10 B 12 C 25 D 125
GATE CSE 2019   Engineering Mathematics
Question 8 Explanation:
 Question 9
Let X be a square matrix. Consider the following two statements on X.

I. X is invertible
II. Determinant of X is non-zero

Which one of the following is TRUE?
 A I implies II; II does not imply I B II implies I; I does not imply II C I does not imply II; II does not imply I D I and II are equivalent statements
GATE CSE 2019   Engineering Mathematics
Question 9 Explanation:
 Question 10
Consider a matrix P whose only eigenvectors are the multiples of $\begin{bmatrix} 1\\ 4 \end{bmatrix}$.
Consider the following statements.
(I) P does not have an inverse
(II) P has a repeated eigenvalue
(III) P cannot be diagonalized
Which one of the following options is correct?
 A Only I and III are necessarily true B Only II is necessarily true C Only I and II are necessarily true D Only II and III are necessarily true
GATE CSE 2018   Engineering Mathematics
Question 10 Explanation:
There are 10 questions to complete.

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