41. The matrix P is the inverse of a matrix Q. If $$I$$ denotes the identity matrix, which one of the following options is correct?
42. A square matrix B is skew-symmetric if
43. Given that, \[{\text{A}} = \left[ {\begin{array}{*{20}{c}}
{ - 5}&{ - 3} \\
2&0
\end{array}} \right]\] and \[I = \left[ {\begin{array}{*{20}{c}}
1&0 \\
0&1
\end{array}} \right],\] the value A3 is
44. Consider the matrices X(4 × 3), Y(4 × 3) and P(2 × 3). The order of [P(XTY)-1 PT]T will be
45. The rank of the matrix \[\left[ {\begin{array}{*{20}{c}}
{ - 4}&1&{ - 1} \\
{ - 1}&{ - 1}&{ - 1} \\
7&{ - 3}&1
\end{array}} \right]\] is
46. Consider the following linear system.
x + 2y - 3z = a
2x + 3y + 3z = b
5x + 9y - 6z = c
This system is consistent if a, b and c satisfy the equation
x + 2y - 3z = a
2x + 3y + 3z = b
5x + 9y - 6z = c
This system is consistent if a, b and c satisfy the equation
47. Consider a 3 × 3 real symmetric matrix S such that two of its eigen values are a ≠ 0, b ≠ 0 with respective eigen vectors \[\left[ {\begin{array}{*{20}{c}}
{{{\text{x}}_1}} \\
{{{\text{x}}_2}} \\
{{{\text{x}}_3}}
\end{array}} \right],\left[ {\begin{array}{*{20}{c}}
{{{\text{y}}_1}} \\
{{{\text{y}}_2}} \\
{{{\text{y}}_3}}
\end{array}} \right].\] If a ≠ b then x1y1 + x2y2 + x3y3 equals
48. For which value of x will the matrix given below become singular?
\[\left[ {\begin{array}{*{20}{c}}
8&{\text{x}}&0 \\
4&0&2 \\
{12}&6&0
\end{array}} \right]\]
\[\left[ {\begin{array}{*{20}{c}} 8&{\text{x}}&0 \\ 4&0&2 \\ {12}&6&0 \end{array}} \right]\]
49. The number of solutions of the simultaneous algebraic equation y = 3x + 3 and y = 3x + 5 is:
50. Given Matrix \[\left[ {\text{A}} \right] = \left[ {\begin{array}{*{20}{c}}
4&2&1&3 \\
6&3&4&7 \\
2&1&0&1
\end{array}} \right],\] the rank of the matrix is
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