Q141
Open questionThe two vectors [1 1 1] and [1, a, a2], where , are
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Q142
Open questionMultiplication of matrices E and F is G. Matrices E and G are {\text{E}} \equiv \left[ {\begin{array}{*{20}{c}}
{\cos \theta }&{ - \sin \theta }&0 \\
{\sin \theta }&{\cos \theta }&0 \\
0&0&1
\end{array}} \right]{\text{and G}} \equiv \left[ {\begin{array}{*{20}{c}}
1&0&0 \\
0&1&0 \\
0&0&1
\end{array}} \right].
What is the matrix F?
What is the matrix F?
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Q143
Open questionIf a matrix {\text{A}} = \left[ {\begin{array}{*{20}{c}}
2&4 \\
1&3
\end{array}} \right] and matrix {\text{B}} = \left[ {\begin{array}{*{20}{c}}
4&6 \\
5&9
\end{array}} \right] the transpose of product of these two matrices i.e., (AB)T is
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Q144
Open questionConsider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigen values of this matrix are -1 and 7. What are the values of a and b?
{\text{A}} = \left( {\begin{array}{*{20}{c}} 1&4 \\ {\text{b}}&{\text{a}} \end{array}} \right)
{\text{A}} = \left( {\begin{array}{*{20}{c}} 1&4 \\ {\text{b}}&{\text{a}} \end{array}} \right)
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Q145
Open questionConsider the following system of equations in three real variables x1, x2 and x3
2x1 - x2 + 3x3 = 1
3x1 - 2x2 + 5x3 = 2
-x1 - 4x2 + x3 = 3
This system of equations has
2x1 - x2 + 3x3 = 1
3x1 - 2x2 + 5x3 = 2
-x1 - 4x2 + x3 = 3
This system of equations has
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Q146
Open questionConsider the matrix
{\text{P}} = \left[ {\begin{array}{*{20}{c}} 1&1&0 \\ 0&1&1 \\ 0&0&1 \end{array}} \right]
The number of distinct eigen values of P is
{\text{P}} = \left[ {\begin{array}{*{20}{c}} 1&1&0 \\ 0&1&1 \\ 0&0&1 \end{array}} \right]
The number of distinct eigen values of P is
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Q147
Open questionThe system of linear equations
4x + 2y = 7
2x + y = 6
has
4x + 2y = 7
2x + y = 6
has
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Q148
Open questionFor what value of a, if any, will the following system of equations in x, y and z have a solution?
2x + 3y = 4; x + y + z = 4; x + 2y - z = a
2x + 3y = 4; x + y + z = 4; x + 2y - z = a
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Q149
Open questionThe eigen value of the following matrix \left[ {\begin{array}{*{20}{c}}
{10}&{ - 4} \\
{18}&{ - 12}
\end{array}} \right]
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Q150
Open questionGiven an orthogonal matrix {\text{A}} = \left[ {\begin{array}{*{20}{c}}
1&1&1&1 \\
1&1&{ - 1}&{ - 1} \\
1&{ - 1}&0&0 \\
0&0&1&{ - 1}
\end{array}} \right],\,{\left[ {{\text{A}}{{\text{A}}^{\text{T}}}} \right]^{ - 1}}\,{\text{is}}
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Q151
Open questionWhich one of the following statements is true for all real symmetric matrices?
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Q152
Open questionx + 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
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Q153
Open questionLet 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
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Q154
Open questionConsider 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?
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Q155
Open questionConsider 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 , 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 , does this system of equations have infinitely many solutions?
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Q156
Open questionM is a 2 × 2 matrix with eigen values 4 and 9. The eigen values of M2 are
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Q157
Open questionThe product of matrices (PQ)-1P is
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Q158
Open questionLet A be a 4 × 3 real matrix with rank 2. Which one of the following statement is TRUE?
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Q159
Open questionThe inverse of matrix \left[ {\begin{array}{*{20}{c}}
0&1&0 \\
1&0&0 \\
0&0&1
\end{array}} \right] is
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Q160
Open questionThe trace and determinant of a 2 × 2 matrix are known to be -2 and -35 respectively. It eigen values are
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