Linear Algebra
Q21.
The larger of the two eigenvalues of the matrix \begin{bmatrix} 4 & 5\\ 2&1 \end{bmatrix} is _______.Q22.
Two eigen values of a 3x3 real matrix P are (2+\sqrt{-1}) and 3.The determinantof P is __________.Q23.
If the following system has non-trivial solution, px+qy+rz=0 qx+ry+pz=0 rx+py+qz=0, then which one of the following options is TRUE?Q25.
Consider the following 2x2 matrix A where two elements are unknown and are marked by a and b. The eigenvalues of this matrix are -1 and 7. What are the values of a and b? A=\begin{pmatrix} 1 & 4\\ b&a \end{pmatrix}.Q26.
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 ____ .Q27.
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___________.Q28.
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________.Q29.
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 ]);Q30.
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_______.