61. Fora given matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}}
2&{ - 2}&3 \\
{ - 2}&{ - 1}&6 \\
1&2&0
\end{array}} \right],\] one of the eigen values is 3. The other two eigen values are
62. Let A be n × n real valued square symmetric matrix of rank 2 with \[\sum\limits_{{\text{i}} = 1}^{\text{n}} {\sum\limits_{{\text{j}} = 1}^{\text{n}} {{\text{A}}_{{\text{ij}}}^2} } = 50.\] Consider the following statements.
I. One eigen value must be in [-5, 5]
II. The eigen value with the largest magnitude must be strictly greater than 5.
Which of the above statements about eigen values of A is/are necessarily CORRECT?
I. One eigen value must be in [-5, 5]
II. The eigen value with the largest magnitude must be strictly greater than 5.
Which of the above statements about eigen values of A is/are necessarily CORRECT?
63. Real matrices [A]3×1, [B]3×3, [C]3×5, [D]5×3, [E]5×5 and [F]5×1 are given. Matrices [B] and [E] are symmetric.
Following statements are made with respect to these matrices.
1. Matrix product [F]T[C]T[B] [C] [F] is a scalar.
2. Matrix product [D]T[F] [D] is always symmetric.
With reference to above statements, which of the following applies?
Following statements are made with respect to these matrices.
1. Matrix product [F]T[C]T[B] [C] [F] is a scalar.
2. Matrix product [D]T[F] [D] is always symmetric.
With reference to above statements, which of the following applies?
64. Let the Eigen vector of the matrix \[\left[ {\begin{array}{*{20}{c}}
1&2 \\
0&2
\end{array}} \right]\] be written in the form \[\left[ {\begin{array}{*{20}{c}}
1 \\
{\text{a}}
\end{array}} \right]\] and \[\left[ {\begin{array}{*{20}{c}}
1 \\
{\text{b}}
\end{array}} \right]\]. What is the value of (a + b) = ?
65. Which one of the following matrices is singular?
66. Let c1 ..... cn be scalars, not all zero, such that \[\sum\limits_{{\text{i}} = 1}^{\text{n}} {{{\text{c}}_{\text{i}}}{{\text{a}}_{\text{i}}}} = 0\] where ai are column vectors in Rn. Consider the set of linear equations Ax = b where A = [a1 ..... an] and \[{\text{b}} = \sum\limits_{{\text{i}} = 1}^{\text{n}} {{{\text{a}}_{\text{i}}}} .\] The set of equations has
67. Two matrices A and B are given below:
\[{\text{A}} = \left[ {\begin{array}{*{20}{c}}
{\text{p}}&{\text{q}} \\
{\text{r}}&{\text{s}}
\end{array}} \right]{\text{;}}\,{\text{B}} = \left[ {\begin{array}{*{20}{c}}
{{{\text{p}}^2} + {{\text{q}}^2}}&{{\text{pr}} + {\text{qs}}} \\
{{\text{pr}} + {\text{qs}}}&{{{\text{r}}^2} + {{\text{s}}^2}}
\end{array}} \right]\]
If the rank of matrix A is N, then the rank of matrix B is
\[{\text{A}} = \left[ {\begin{array}{*{20}{c}} {\text{p}}&{\text{q}} \\ {\text{r}}&{\text{s}} \end{array}} \right]{\text{;}}\,{\text{B}} = \left[ {\begin{array}{*{20}{c}} {{{\text{p}}^2} + {{\text{q}}^2}}&{{\text{pr}} + {\text{qs}}} \\ {{\text{pr}} + {\text{qs}}}&{{{\text{r}}^2} + {{\text{s}}^2}} \end{array}} \right]\]
If the rank of matrix A is N, then the rank of matrix B is
68. Consider the matrix \[{\text{P}} = \left[ {\begin{array}{*{20}{c}}
{\frac{1}{{\sqrt 2 }}}&0&{\frac{1}{{\sqrt 2 }}} \\
0&1&0 \\
{\frac{{ - 1}}{{\sqrt 2 }}}&0&{\frac{1}{{\sqrt 2 }}}
\end{array}} \right]\]
Which one of the following statements about P is INCORRECT?
Which one of the following statements about P is INCORRECT?
69. The eigen values and the corresponding eigen vectors of a 2 × 2 matrix are given by
\[\begin{array}{*{20}{c}}
{{\text{Eigen value}}}&{{\text{Eigen vector}}} \\
{{\lambda _1} = 8}&{{{\text{v}}_1} = \left[ {\begin{array}{*{20}{c}}
1 \\
1
\end{array}} \right]} \\
{{\lambda _2} = 4}&{{{\text{v}}_2} = \left[ {\begin{array}{*{20}{c}}
1 \\
{ - 1}
\end{array}} \right]}
\end{array}\]
The matrix is
\[\begin{array}{*{20}{c}} {{\text{Eigen value}}}&{{\text{Eigen vector}}} \\ {{\lambda _1} = 8}&{{{\text{v}}_1} = \left[ {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right]} \\ {{\lambda _2} = 4}&{{{\text{v}}_2} = \left[ {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right]} \end{array}\]
The matrix is
70. A scalar valued function is defined as f(X) = XTAX + bTX + c, where A is a symmetric positive definite matrix with dimension n × n; b and x are vectors of dimension n × 1. The minimum value of f(X) will occur when X equals
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