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
\begin{pmatrix} -9 &-6 &-2 &-4 \\ -8& -6 & -3 & -1 \\ 20 & 15 & 8 & 5 \\ 32& 21& 7&12 \end{pmatrix}
MSQ
\begin{pmatrix}
-1\\
1\\
0\\
1
\end{pmatrix} | |
\begin{pmatrix}
1\\
0\\
-1\\
0
\end{pmatrix} | |
\begin{pmatrix}
-1\\
0\\
2\\
2
\end{pmatrix} | |
\begin{pmatrix}
0\\
1\\
-3\\
0
\end{pmatrix} |
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?
\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?
L_{32}=2,U_{33}=-\frac{1}{2},x_1=-1 | |
L_{32}=2,U_{33}=2,x_1=-1 | |
L_{32}=-\frac{1}{2},U_{33}=2,x_1=0 | |
L_{32}=-\frac{1}{2},U_{33}=-\frac{1}{2},x_1=0 |
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)
wheretr() represents the trace of a matrix. Which one of the following holds?
Statement 1: tr(AB) = tr(BA)
Statement 2: tr(CD) = tr(DC)
wheretr() represents the trace of a matrix. Which one of the following holds?
Statement 1 is correct and Statement 2 is wrong. | |
Statement 1 is wrong and Statement 2 is correct. | |
Both Statement 1 and Statement 2 are correct. | |
Both Statement 1 and Statement 2 are wrong. |
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 _______
1 | |
2 | |
3 | |
4 |
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 __________.
\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 __________.
1 | |
3 | |
4 | |
6 |
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
8 | |
16 | |
18 | |
20 |
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?
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?
I and II only | |
I and IV only | |
II and III only | |
III and IV only |
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 _________ .
\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 _________ .
10 | |
12 | |
25 | |
125 |
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?
I. X is invertible
II. Determinant of X is non-zero
Which one of the following is TRUE?
I implies II; II does not imply I | |
II implies I; I does not imply II | |
I does not imply II; II does not imply I | |
I and II are equivalent statements |
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?
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?
Only I and III are necessarily true | |
Only II is necessarily true | |
Only I and II are necessarily true | |
Only II and III are necessarily true |
Question 10 Explanation:
There are 10 questions to complete.
Please make a login portal, so we can track our progress
Dear Chinmay Rajpurohit,
We will consider your suggestion.
Agree
Agreed
Hey Team,
Linear Algebra
In Q1 from Gate 2022 Option C should be [-1 0 2 2]ᵀ
Option “C” given in Gate 2022 for MSQ is wrong it should be {-1 0 2 2}^T Kindly Look Into it.
Dear Raghav,
We have updated the option.
Thank You.