Linear Algebra
Q31.
The product of the non-zero eigenvalues of the matrix \begin{bmatrix} 1 & 0&0 & 0&1 \\ 0& 1& 1 & 1 & 0\\ 0& 1& 1& 1&0 \\ 0 & 1 & 1 & 1 & 0\\ 1&0 & 0&0 & 1 \end{bmatrix} is_______.Q32.
If V1 and V2 are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dimension of V1\capV2 is _______.Q33.
A non-zero polynomial f(x) of degree 3 has roots at x = 1,x = 2 and x = 3. Which one of the following must be TRUE?Q34.
The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4-by-4 symmetric positive definite matrix isQ35.
What is the matrix transformation which takes the independent vectors \begin{pmatrix} 1 \\ 2 \end{pmatrix}\text{ and }\begin{pmatrix} 2 \\ 5 \end{pmatrix} and transforms them to \begin{pmatrix} 1\\ 1 \end{pmatrix} \text{ and }\begin{pmatrix} 3 \\ 2 \end{pmatrix} respectively?Q36.
Consider the following system of equations: 3x + 2y = 1 4x + 7z = 1 x + y + z = 3 x - 2y + 7z = 0 The number of solutions for this system isQ37.
If the matrix A is such that \begin{bmatrix} 2\\ -4\\ 7 \end{bmatrix}\begin{bmatrix} 1 &9 &5 \end{bmatrix} Then the determinant of A is equal to ________.Q38.
Consider the matrix as given below. \begin{bmatrix} 1 & 2&3 \\ 0& 4 & 7\\ 0&0 & 3 \end{bmatrix} Which one of the following provides the CORRECT values of eigenvalues of the matrix?Q40.
Which one of the following does NOT equal \begin{bmatrix} 1 & x&x^{2} \\ 1& y & y^{2}\\ 1&z & z^{2} \end{bmatrix}?