Marks 1
Consider a system of linear equations Ax = b, where
$$A = \left[ {\matrix{
1 \hfill & { - \sqrt 2 } \hfill & 3 \hfill \cr
{ - 1} \hfill & {\sq...
Let $$\alpha$$, $$\beta$$ two non-zero real numbers and v1, v2 be two non-zero real vectors of size 3 $$\times$$ 1. Suppose that v1 and v2 satisfy $$v...
Let
M
be a real
4 $$ \times $$ 4
matrix. Consider the following statements :
S1:
M
has
4
linearly independent eigenvectors.
S2:
M
has
4
distinct eige...
Consider matrix $$A = \left[ {\matrix{
k & {2k} \cr
{{k^2} - k} & {{k^2}} \cr
} } \right]$$ and
vector $$X = \left[ {\matrix{
{...
The rank of the matrix $$M = \left[ {\matrix{
5 & {10} & {10} \cr
1 & 0 & 2 \cr
3 & 6 & 6 \cr
} } \right]$$ i...
Consider the $$5 \times 5$$ matrix $$A = \left[ {\matrix{
1 & 2 & 3 & 4 & 5 \cr
5 & 1 & 2 & 3 & 4 \cr
4 &...
Consider a $$2 \times 2$$ square matrix $$A = \left[ {\matrix{
\sigma & x \cr
\omega & \sigma \cr
} } \right]$$
Where $$x$$ is ...
The value of $$x$$ for which the matrix $$A = \left[ {\matrix{
3 & 2 & 4 \cr
9 & 7 & {13} \cr
{ - 6} & { - 4} & {...
Let $${M^4} = {\rm I}$$ (where $${\rm I}$$ denotes the identity matrix) and $$M \ne {\rm I},\,\,{M^2} \ne {\rm I}$$ and $${M^3} \ne {\rm I}$$. Then, ...
For $$A = \left[ {\matrix{
1 & {\tan x} \cr
{ - \tan x} & 1 \cr
} } \right],$$ the determinant of $${A^T}\,{A^{ - 1}}$$ is
The value of $$'x'$$ for which all the eigenvalues of the matrix given below are real is $$\left[ {\matrix{
{10} & {5 + j} & 4 \cr
x ...
Consider system of linear equations :
$$$x-2y+3z=-1$$$
$$$x-3y+4z=1$$$ and
$$$-2x+4y-6z=k,$$$
The value of $$'k'$$ for which the system has infinite...
The value of $$'P'$$ such that the vector $$\left[ {\matrix{
1 \cr
2 \cr
3 \cr
} } \right]$$ is an eigenvector of the matrix $$\left...
Which one of the following statements is NOT true for a square matrix $$A$$?
The system of linear equations $$\left( {\matrix{
2 & 1 & 3 \cr
3 & 0 & 1 \cr
1 & 2 & 5 \cr
} } \right)\left(...
The determinant of matrix $$A$$ is $$5$$ and the determinant of matrix $$B$$ is $$40.$$ The determinant of matrix $$AB$$ is _______.
The maximum value of the determinant among all $$2 \times 2$$ real symmetric matrices with trace $$14$$ is ______.
For matrices of same dimension $$M,N$$ and scalar $$c,$$ which one of these properties DOES NOT ALWAYS hold ?
Consider the matrix $${J_6} = \left[ {\matrix{
0 & 0 & 0 & 0 & 0 & 1 \cr
0 & 0 & 0 & 0 & 1 & 0 \cr
...
$$A$$ real $$\left( {4\,\, \times \,\,4} \right)$$ matrix $$A$$ satisfies the equation $${A^2} = {\rm I},$$ where $${\rm I}$$ is the $$\left( {4\,\, \...
The minimum eigenvalue of the following matrix is $$\left[ {\matrix{
3 & 5 & 2 \cr
5 & {12} & 7 \cr
2 & 7 & 5 \c...
Let $$A$$ be an $$m\,\, \times \,\,n$$ matrix and $$B$$ an $$n\,\, \times \,\,m$$ matrix. It is given that determinant $$\left( {{{\rm I}_m} + AB} \ri...
Given that $$A = \left[ {\matrix{
{ - 5} & { - 3} \cr
2 & 0 \cr
} } \right]$$ and $${\rm I} = \left[ {\matrix{
1 & 0 \cr
...
The system of equations $$x+y+z=6,$$ $$x+4y+6z=20,$$ $$x + 4y + \lambda z = \mu $$ has no solution for values of $$\lambda $$ and $$\mu $$ given by
The eigen values of a skew-symmetric matrix are
All the four entries of $$2$$ $$x$$ $$2$$ matrix
$$P = \left[ {\matrix{
{{p_{11}}} & {{p_{12}}} \cr
{{p_{21}}} & {{p_{22}}} \cr
}...
The system of linear equations $$\left. {\matrix{
{4x + 2y = 7} \cr
{2x + y = 6} \cr
} } \right\}$$ has
For the matrix $$\left[ {\matrix{
4 & 2 \cr
2 & 4 \cr
} } \right].$$ The eigen value corresponding to the eigen vector $$\left[ {\...
The rank of the matrix $$\left[ {\matrix{
1 & 1 & 1 \cr
1 & { - 1} & 0 \cr
1 & 1 & 1 \cr
} } \right]$$ is
The eigen values of the matrix $$\left[ {\matrix{
2 & { - 1} & 0 & 0 \cr
0 & 3 & 0 & 0 \cr
0 & 0 & { - 2}...
The eigen values of the matrix $$A = \left[ {\matrix{
0 & 1 \cr
1 & 0 \cr
} } \right]$$ are
The following system of equations
$${{x_1} + {x_2} + {x_3} = 3}$$
$${{x_1} - {x_3} = 0}$$
$${{x_1} - {x_2} + {x_3} = 1}$$ has
The rank of $$\left( {m \times n} \right)$$ matrix $$\left( {m < n} \right)$$ cannot be more then
Marks 2
The rank of the matrix $$\left[ {\matrix{
1 & { - 1} & 0 & 0 & 0 \cr
0 & 0 & 1 & { - 1} & 0 \cr
0 & 1...
If the vectors $${e_1} = \left( {1,0,2} \right),\,{e_2} = \left( {0,1,0} \right)$$ and $${e_3} = \left( { - 2,0,1} \right)$$ form an orthogonal basis ...
The matrix $$A = \left[ {\matrix{
a & 0 & 3 & 7 \cr
2 & 5 & 1 & 3 \cr
0 & 0 & 2 & 4 \cr
0 & ...
A sequence $$x\left[ n \right]$$ is specified as
$$$\left[ {\matrix{
{x\left[ n \right]} \cr
{x\left[ {n - 1} \right]} \cr
} } \right] = ...
The eigen values of the following matrix $$\left[ {\matrix{
{ - 1} & 3 & 5 \cr
{ - 3} & { - 1} & 6 \cr
0 & 0 & 3 ...
The eigen values and the correspondinng eigen vectors of a $$2 \times 2$$ matrix are given by
Eigen value
$${\lambda _1} = 8$$
$${\lambda _2} = 4$$
E...
Given the matrix $$\left[ {\matrix{
{ - 4} & 2 \cr
4 & 3 \cr
} } \right],$$ the eigen vector is
Given an orthogonal matrix $$A = \left[ {\matrix{
1 & 1 & 1 & 1 \cr
1 & 1 & { - 1} & { - 1} \cr
1 & { - 1} &a...
If $$A = \left[ {\matrix{
2 & { - 0.1} \cr
0 & 3 \cr
} } \right]$$ and $${A^{ - 1}} = \left[ {\matrix{
{{\raise0.5ex\hbox{$\scr...