51. Which one of the following statements is true for all real symmetric matrices?
52. x + 2y + z = 4
2x + y + 2z = 5
x - y + z = 1
The system of algebraic given below has
2x + y + 2z = 5
x - y + z = 1
The system of algebraic given below has
53. Let P ≠ 0 be a 3 × 3 real matrix. There exist linearly independent vectors x and y such that PX = 0 and PY = 0. The dimension to the range space of P is
54. Consider the systems, each consisting of m linear equations in n variables.
I. If m < n, then all such systems have a solution.
II. If m > 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?
I. If m < n, then all such systems have a solution.
II. If m > 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?
55. Consider the following system of linear equations
\[\left[ {\begin{array}{*{20}{c}}
2&1&{ - 4} \\
4&3&{ - 12} \\
1&2&{ - 8}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{\text{x}} \\
{\text{y}} \\
{\text{z}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
\alpha \\
5 \\
7
\end{array}} \right]\]
Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of \[\alpha \], does this system of equations have infinitely many solutions?
\[\left[ {\begin{array}{*{20}{c}} 2&1&{ - 4} \\ 4&3&{ - 12} \\ 1&2&{ - 8} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\text{x}} \\ {\text{y}} \\ {\text{z}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} \alpha \\ 5 \\ 7 \end{array}} \right]\]
Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of \[\alpha \], does this system of equations have infinitely many solutions?
56. M is a 2 × 2 matrix with eigen values 4 and 9. The eigen values of M2 are
57. The product of matrices (PQ)-1P is
58. Let A be a 4 × 3 real matrix with rank 2. Which one of the following statement is TRUE?
59. The inverse of matrix \[\left[ {\begin{array}{*{20}{c}}
0&1&0 \\
1&0&0 \\
0&0&1
\end{array}} \right]\] is
60. The trace and determinant of a 2 × 2 matrix are known to be -2 and -35 respectively. It eigen values are
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