# Linear Algebra

 Question 1
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 1 Explanation:
 Question 2
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 2 Explanation:
 Question 3
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 3 Explanation:
 Question 4
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 4 Explanation:
 Question 5
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 5 Explanation:
 Question 6
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 6 Explanation:
 Question 7
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 7 Explanation:
 Question 8
Consider a matrix $A=uv^{T}\; where \; u=\begin{bmatrix} 1\\ 2 \end{bmatrix},v=\begin{bmatrix} 1\\ 1 \end{bmatrix}$ Note that $v^{T}$ denotes the transpose of v. The largest eigenvalue of A is _____.
 A 3 B 2 C 1 D 4
GATE CSE 2018   Engineering Mathematics
Question 8 Explanation:
 Question 9
If A is a skew symmetric matrix then $A^{t}$ is
 A Diagonal matrix B A C 0 D -A
ISRO CSE 2017   Engineering Mathematics
Question 9 Explanation:
 Question 10
If the characteristics polynomial of 3x3 matrix M over R ( the set of real numbers) is $\lambda ^{3}-4\lambda ^{2}+a\lambda +30,a\in R$, and one eigenvalue of M is 2, then the largest among the absolute values of the eigenvalues of M is ________.
 A 2 B 3 C 4 D 5
GATE CSE 2017 SET-2   Engineering Mathematics
Question 10 Explanation:
There are 10 questions to complete.

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