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 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 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 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 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 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 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 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 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 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
There are 10 questions to complete.

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