Linear Algebra


Question 1
Let A be the adjacency matrix of the graph with vertices {1, 2, 3, 4, 5}.

Let \lambda _1,\lambda _2,\lambda _3,\lambda _4,\; and \; \lambda _5 be the five eigenvalues of A. Note that these eigenvalues need not be distinct.
The value of \lambda _1+\lambda _2+\lambda _3+ \lambda _4+ \lambda _5 = _____
A
1
B
2
C
3
D
4
GATE CSE 2023   Engineering Mathematics
Question 2
Let A=\begin{bmatrix} 1 & 2 & 3 &4 \\ 4& 1& 2 &3 \\ 3& 4 & 1 &2 \\ 2 &3 &4 &1 \end{bmatrix} and B=\begin{bmatrix} 3& 4 & 1 &2 \\ 4& 1& 2 &3 \\ 1 & 2 & 3 &4 \\ 2 &3 &4 &1 \end{bmatrix}
Let det(A) and det(B) denote the determinants of the matrices A and B, respectively.
Which one of the options given below is TRUE?
A
det(A) = det(B)
B
det(B) = - det(A)
C
det(A)=0
D
det(AB) = det(A) + det(B)
GATE CSE 2023   Engineering Mathematics


Question 3
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\\ 2\\ 2 \end{pmatrix}
D
\begin{pmatrix} 0\\ 1\\ -3\\ 0 \end{pmatrix}
GATE CSE 2022   Engineering Mathematics
Question 4
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 5
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?
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




There are 5 questions to complete.

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