Engineering Mathematics
Q51.
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 _________ .Q52.
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}MSQQ53.
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 toQ55.
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 _____.Q56.
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 ________.Q57.
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?Q58.
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?Q59.
Let P=\begin{bmatrix} 1 & 1&-1 \\ 2&-3 & 4\\ 3 &-2 & 3 \end{bmatrix} and Q=\begin{bmatrix} -1 & -2&-1 \\ 6 & 12& 6\\ 5&10 & 5 \end{bmatrix} be two matrices. Then the rank of P +Q is _____________.Q60.
Let c_{1}....c_{n} be scalars, not all zero, such that \sum_{i=1}^{n}c_{i}a_{i}=0 where a_{i} are column vectors in R^{n}. Consider the set of linear equations Ax = b where A=a_{1}....a_{n} and b=\sum_{i=1}^{n}a_{i}. The set of equations has