Linear Algebra
Q1.
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?Q2.
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 = _____Q3.
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?Q4.
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 _______Q5.
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 asL= \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?Q6.
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 _________ .Q7.
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 toQ8.
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}MSQQ9.
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 ________.