# 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 1 Explanation:
 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 2 Explanation:

 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 3 Explanation:
 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 4 Explanation:
 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: $tr(CD) = tr(DC)$

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 5 Explanation:

There are 5 questions to complete.

### 16 thoughts on “Linear Algebra”

1. Please make a login portal, so we can track our progress

• Dear Chinmay Rajpurohit,

• Agree

2. Hey Team,
Linear Algebra
In Q1 from Gate 2022 Option C should be [-1 0 2 2]ᵀ

3. 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.

4. Question no. 74
Value of matrices a13 = -4 not -1 , Please verify and correct it\

• Question updated with value -4.

5. Question no. 79 some coding error . please click on correct option and check the error.

6. Question no. 22
Please remove ? marks from its create confusion.