Engineering Mathematics
Q71.
The rank of the matrix A=\left(\begin{array}{cccc} 1 & 2 & 1 & -1 \\ 9 & 5 & 2 & 2 \\ 7 & 1 & 0 & 4 \end{array}\right) is ____ .Q72.
Perform the following operations on the matrix \begin{bmatrix} 3 & 4&45 \\ 7& 9& 105\\ 13&2 & 195 \end{bmatrix}. (i) Add the third row to the second row (ii) Subtract the third column from the first column. The determinant of the resultant matrix is___________.Q73.
In the LU decomposition of the matrix \begin{bmatrix} 2 & 2\\ 4&9 \end{bmatrix}, if the diagonal elements of U are both 1, then the lower diagonal entry l_{22} of L is________.Q74.
Let A be a square matrix size n x n. Consider the following pseudocode. What is the expected output? C = 100; for i = 1 to n do for j = 1 to n do { Temp = A[ i ] [ j ] + C ; A [ i ] [ j ] = A [ j ] [ i ] ; A [ j ] [ i ] = Temp - C ; } for i = 1 to n do for j = 1 to n do output (A[ i ] [ j ]);Q75.
Which one of the following statements is TRUE about every n x n matrix with only real eigenvalues?Q76.
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_______.Q77.
If V1 and V2 are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dimension of V1\capV2 is _______.Q78.
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?Q79.
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 isQ80.
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?