Linear Algebra
Q11.
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 ________.Q12.
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?Q13.
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?Q14.
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 _____________.Q15.
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 hasQ16.
Let A be nxn real valued square symmetric matrix of rank 2 with \sum_{i=1}^{n}\sum_{j=1}^{n}A^{2}_{ij}=50. Consider the following statements. (I) One eigen value must be in [-5, 5] (II) The eigen value with the largest magnitude must be strictly greater than 5. Which of the above statements about eigen values of A is/are necessarily CORRECT?Q17.
Consider a quadratic equation x^{2} -13x +36 = 0 with coefficients in a base b. The solutions of this equation in the same base b are x = 5 and x = 6. Then b = ___________.Q18.
Consider the systems,each consisting of m linear equations in n variables. I. If m \lt n, then all such systems have a solution II. If m \gt n, then none of these systems has a solution III. If m = n, then there exists a system which has a solution Which one of the following is CORRECT?Q19.
Suppose that the eigen values of matrix A are 1, 2, 4. The determinant of (A^{-1})^{T} is _________.Q20.
In the given matrix \begin{bmatrix} 1 & -1&2 \\ 0& 1 & 0\\ 1&2 & 1 \end{bmatrix}, one of the eigenvalues is 1. The eigenvectors corresponding to the eigenvalue 1 are