Matrices and Determinants

The greatest value of c \(\in\) R for which the system
of linear equations
x – cy – cz = 0
cx – y + cz = 0
cx + cy – z = 0
has a non-trivial solution, is :

Let \(A = \left( {\matrix{ {\cos \alpha } & { - \sin \alpha } \cr {\sin \alpha } & {\cos \alpha } \cr } } \right)\), (\(\alpha\) \(\in\) R) such that $${A^{32}} = \left( {\matrix{
0 & { - 1} \cr
1 & 0 \cr

} } \rig...

Let the number 2,b,c be in an A.P. and
A = \(\left[ {\matrix{ 1 & 1 & 1 \cr 2 & b & c \cr 4 & {{b^2}} & {{c^2}} \cr } } \right]\). If det(A) \(\in\) [2, 16], then c
lies in the interval :

Let P = \(\left[ {\matrix{ 1 & 0 & 0 \cr 3 & 1 & 0 \cr 9 & 3 & 1 \cr } } \right]\) and Q = [qij] be two 3 \(\times\) 3 matrices such that Q – P5 = I3.
Then \({{{q_{21}} + {q_{31}}} \over {{q_{32}}}}\) is equal to :

An ordered pair (\(\alpha\), \(\beta\)) for which the system of linear equations
(1 + \(\alpha\)) x + \(\beta\)y + z = 2
\(\alpha\)x + (1 + \(\beta\))y + z = 3
\(\alpha\)x + \(\beta\)y + 2z = 2
has a unique solution, is :

The set of all values of \(\lambda\) for which the system of linear equations
x – 2y – 2z = \(\lambda\)x
x + 2y + z = \(\lambda\)y
– x – y = \(\lambda\)z
has a non-trivial solutions :

If   A = \(\left[ {\matrix{ 1 & {\sin \theta } & 1 \cr { - \sin \theta } & 1 & {\sin \theta } \cr { - 1} & { - \sin \theta } & 1 \cr } } \right]\);
then for all \(\theta\) \(\in\) $$\left( {{{3\pi } \over 4},{{5\pi } \o...

If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = \(\left[ {\matrix{ 2 & 3 \cr 5 & { - 1} \cr } } \right]\), then AB is equal
to :

If \(B = \left[ {\matrix{ 5 & {2\alpha } & 1 \cr 0 & 2 & 1 \cr \alpha & 3 & { - 1} \cr } } \right]\) is the inverse of a 3 × 3 matrix A, then the sum of all values of \(\alpha\) for which
det(A) + 1 = 0, is :

A value of \(\theta \in \left( {0,{\pi \over 3}} \right)\), for which
$$\left| {\matrix{
{1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } & {4\cos 6\theta } \cr
{{{\cos }^2}\theta } & {1 + {{\sin }^2}\theta } & {4\cos 6\theta } \c...

Let A = \(\left( {\matrix{ 0 & {2q} & r \cr p & q & { - r} \cr p & { - q} & r \cr } } \right).\)   If  AAT = I3,   then   \(\left| p \right|\) is :

If the system of linear equations
2x + 2y + 3z = a
3x – y + 5z = b
x – 3y + 2z = c
where a, b, c are non zero real numbers, has more one solution, then :

If  \(\left| {\matrix{ {a - b - c} & {2a} & {2a} \cr {2b} & {b - c - a} & {2b} \cr {2c} & {2c} & {c - a - b} \cr } } \right|\)
      = (a + b + c) (x + a + b + c)2, x \(\ne\) 0,
then x is equal to :

Let A and B be two invertible matrices of order 3 \(\times\) 3. If det(ABAT) = 8 and det(AB–1) = 8,
then det (BA–1 BT) is equal to :

If the system of equations
x + y + z = 5
x + 2y + 3z = 9
x + 3y + az = \(\beta\)
has infinitely many solutions, then \(\beta\) \(-\) \(\alpha\) equals -

Let  d \(\in\) R, and 
$$A = \left[ {\matrix{
{ - 2} & {4 + d} & {\left( {\sin \theta } \right) - 2} \cr
1 & {\left( {\sin \theta } \right) + 2} & d \cr
5 & {\left( {2\sin \theta } \right) - d} & {\left( { - \sin \theta } \ri...

Let A = \(\left[ {\matrix{ 2 & b & 1 \cr b & {{b^2} + 1} & b \cr 1 & b & 2 \cr } } \right]\) where b > 0.
Then the minimum value of \({{\det \left( A \right)} \over b}\) is -

The number of values of \(\theta\) \(\in\) (0, \(\pi\)) for which the system of linear equations
x + 3y + 7z = 0
\(-\) x + 4y + 7z = 0
(sin3\(\theta\))x + (cos2\(\theta\))y + 2z = 0.
has a non-trival solution, is -

If the system of linear equations
x + y + z = 5
x + 2y + 2z = 6
x + 3y + \(\lambda\)z = \(\mu\), (\(\lambda\), \(\mu\) \(\in\) R), has infinitely many solutions, then the value of \(\lambda\) + \(\mu\) is :

If \({\Delta _1} = \left| {\matrix{ x & {\sin \theta } & {\cos \theta } \cr { - \sin \theta } & { - x} & 1 \cr {\cos \theta } & 1 & x \cr } } \right|\) and
$${\Delta _2} = \left| {\matrix{
x & {\sin 2\theta } & {\cos 2\...

Let \(\lambda\) be a real number for which the system of linear equations x + y + z = 6, 4x + \(\lambda\)y – \(\lambda\)z = \(\lambda\) – 2,
3x + 2y – 4z = – 5 has infinitely many solutions. Then \(\lambda\) is a root of the quadratic ...

The sum of the real roots of the equation
\(\left| {\matrix{ x & { - 6} & { - 1} \cr 2 & { - 3x} & {x - 3} \cr { - 3} & {2x} & {x + 2} \cr } } \right| = 0\), is equal to :

The number of values of k for which the system of linear equations,
(k + 2)x + 10y = k
kx + (k +3)y = k -1
has no solution, is :

Let A = \(\left[ {\matrix{ 1 & 0 & 0 \cr 1 & 1 & 0 \cr 1 & 1 & 1 \cr } } \right]\) and B = A20. Then the sum of the elements of the first column of B is :

Let \(A\) be a matrix such that \(A.\left[ {\matrix{ 1 & 2 \cr 0 & 3 \cr } } \right]\) is a scalar matrix and |3A| = 108.
Then A2 equals :

Let S be the set of all real values of k for which the systemof linear equations
x + y + z = 2
2x + y \(-\) z = 3
3x + 2y + kz = 4
has a unique solution. Then S is :

If the system of linear equations
x + ay + z = 3
x + 2y + 2z = 6
x + 5y + 3z = b
has no solution, then :

Suppose A is any 3\(\times\) 3 non-singular matrix and ( A \(-\) 3I) (A \(-\) 5I) = O where I = I3 and O = O3. If \(\alpha\)A + \(\beta\)A-1 = 4I, then \(\alpha\) + \(\beta\) is equal to :

If the system of linear equations
x + ky + 3z = 0
3x + ky - 2z = 0
2x + 4y - 3z = 0
has a non-zero solution (x, y, z), then \({{xz} \over {{y^2}}}\) is equal to

If \(\left| {\matrix{ {x - 4} & {2x} & {2x} \cr {2x} & {x - 4} & {2x} \cr {2x} & {2x} & {x - 4} \cr } } \right| = \left( {A + Bx} \right){\left( {x - A} \right)^2}\)
then the ordered pair (A, B) is equal to :

For two 3 × 3 matrices A and B, let A + B = 2BT and 3A + 2B = I3, where BT is
the transpose of B and I3 is 3 × 3 identity matrix. Then :

The number of real values of \(\lambda\) for which the system of linear equations
2x + 4y \(-\) \(\lambda\)z = 0
4x + \(\lambda\)y + 2z = 0
\(\lambda\)x + 2y + 2z = 0
has infinitely many solutions, is :

If
\(S = \left\{ {x \in \left[ {0,2\pi } \right]:\left| {\matrix{ 0 & {\cos x} & { - \sin x} \cr {\sin x} & 0 & {\cos x} \cr {\cos x} & {\sin x} & 0 \cr } } \right| = 0} \right\},\)
then $$\sum\limits_{x \in S} {\tan \left...

Let A be any 3 \(\times\) 3 invertible matrix. Then which one of the following is not always true ?

If \(A = \left[ {\matrix{ 2 & { - 3} \cr { - 4} & 1 \cr } } \right]\),
then adj(3A2 + 12A) is equal to

If S is the set of distinct values of 'b' for which the following system of linear equations
x + y + z = 1
x + ay + z = 1
ax + by + z = 0
has no solution, then S is :

If P = \(\left[ {\matrix{ {{{\sqrt 3 } \over 2}} & {{1 \over 2}} \cr { - {1 \over 2}} & {{{\sqrt 3 } \over 2}} \cr } } \right],A = \left[ {\matrix{ 1 & 1 \cr 0 & 1 \cr } } \right]\,\,\,\)
Q = PAPT, then PT Q2015 P ...

The number of distinct real roots of the equation,
\(\left| {\matrix{ {\cos x} & {\sin x} & {\sin x} \cr {\sin x} & {\cos x} & {\sin x} \cr {\sin x} & {\sin x} & {\cos x} \cr } } \right| = 0\) in the interval $$\left[ { -...

Let A be a 3 \(\times\) 3 matrix such that A2 \(-\) 5A + 7I = 0
Statement - I :  
A\(-\)1 = \({1 \over 7}\) (5I \(-\) A).
Statement - II :
The polynomial A3 \(-\) 2A2 \(-\) 3A + I can be reduced to 5(A \(-\) 4I).
Then :

If    A = \(\left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]\),
then the determinant of the matrix (A2016 − 2A2015 − A2014) is :

The system of linear equations
\(\matrix{ {x + \lambda y - z = 0} \cr {\lambda x - y - z = 0} \cr {x + y - \lambda z = 0} \cr }\)
has a non-trivial solution for :

If \(A = \left[ {\matrix{ {5a} & { - b} \cr 3 & 2 \cr } } \right]\) and \(A\) adj \(A=A\) \({A^T},\) then \(5a+b\) is equal to :

The set of all values of \(\lambda\) for which the system of linear equations:
$$\matrix{
{2{x_1} - 2{x_2} + {x_3} = \lambda {x_1}} \cr
{2{x_1} - 3{x_2} + 2{x_3} = \lambda {x_2}} \cr
{ - {x_1} + 2{x_2} = \lambda {x_3}} \cr

...

If \(A = \left[ {\matrix{ 1 & 2 & 2 \cr 2 & 1 & { - 2} \cr a & 2 & b \cr } } \right]\) is a matrix satisfying the equation
\(A{A^T} = 9\text{I},\) where \(I\) is \(3 \times 3\) identity matrix, then the ordered
pair $$(a,...

If \(\alpha ,\beta \ne 0,\) and \(f\left( n \right) = {\alpha ^n} + {\beta ^n}\) and
$$$\left| {\matrix{
3 & {1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} \cr
{1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} & {1 + f\left( 3 ...

If \(A\) is a \(3 \times 3\) non-singular matrix such that \(AA'=A'A\) and
\(B = {A^{ - 1}}A',\) then \(BB'\) equals:

If \(P = \left[ {\matrix{ 1 & \alpha & 3 \cr 1 & 3 & 3 \cr 2 & 4 & 4 \cr } } \right]\) is the adjoint of a \(3 \times 3\) matrix \(A\) and
\(\left| A \right| = 4,\) then \(\alpha\) is equal to :

The number of values of \(k\), for which the system of equations : \($\matrix{ {\left( {k + 1} \right)x + 8y = 4k} \cr {kx + \left( {k + 3} \right)y = 3k - 1} \cr }\)$
has no solution, is

Let \(A = \left( {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 0 \cr 3 & 2 & 1 \cr } } \right)\). If \({u_1}\) and \({u_2}\) are column matrices such
that \(A{u_1} = \left( {\matrix{ 1 \cr 0 \cr 0 \cr } } \right)\) and...

Let \(P\) and \(Q\) be \(3 \times 3\) matrices \(P \ne Q.\) If \({P^3} = {Q^3}\) and
\({P^2}Q = {Q^2}P\) then determinant of \(\left( {{P^2} + {Q^2}} \right)\) is equal to :