Matrices and Determinants

Let I be an identity matrix of order 2 \(\times\) 2 and P = \(\left[ {\matrix{ 2 & { - 1} \cr 5 & { - 3} \cr } } \right]\). Then the value of n\(\in\)N for which Pn = 5I \(-\) 8P is equal to ____________.

The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k2 has no solution if k is equal to :

If \(A = \left( {\matrix{ 0 & {\sin \alpha } \cr {\sin \alpha } & 0 \cr } } \right)\) and \(\det \left( {{A^2} - {1 \over 2}I} \right) = 0\), then a possible value of \(\alpha\) is :

If \(A = \left[ {\matrix{ 2 & 3 \cr 0 & { - 1} \cr } } \right]\), then the value of det(A4) + det(A10 \(-\) (Adj(2A))10) is equal to _____________.

Let \(A = \left[ {\matrix{ a & b \cr c & d \cr } } \right]\) and \(B = \left[ {\matrix{ \alpha \cr \beta \cr } } \right] \ne \left[ {\matrix{ 0 \cr 0 \cr } } \right]\) such that AB = B and a + d = 2021,...

If 1, log10(4x \(-\) 2) and log10\(\left( {{4^x} + {{18} \over 5}} \right)\) are in arithmetic progression for a real number x, then the value of the determinant $$\left| {\matrix{
{2\left( {x - {1 \over 2}} \right)} & {x - 1} & {{x^2}} ...

If x, y, z are in arithmetic progression with common difference d, x \(\ne\) 3d, and the determinant of the matrix \(\left[ {\matrix{ 3 & {4\sqrt 2 } & x \cr 4 & {5\sqrt 2 } & y \cr 5 & k & z \cr } } \right]\) is zero, then...

Let \(P = \left[ {\matrix{ { - 30} & {20} & {56} \cr {90} & {140} & {112} \cr {120} & {60} & {14} \cr } } \right]\) and $$A = \left[ {\matrix{
2 & 7 & {{\omega ^2}} \cr
{ - 1} & { - \omega } & 1 \cr
0 & { - \om...

The total number of 3 \(\times\) 3 matrices A having entries from the set {0, 1, 2, 3} such that the sum of all the diagonal entries of AAT is 9, is equal to _____________.

Let \(A = \left[ {\matrix{ i & { - i} \cr { - i} & i \cr } } \right],i = \sqrt { - 1}\). Then, the system of linear equations $${A^8}\left[ {\matrix{
x \cr
y \cr

} } \right] = \left[ {\matrix{
8 \cr
{64} \c...

Let \(A = \left[ {\matrix{ {{a_1}} \cr {{a_2}} \cr } } \right]\) and \(B = \left[ {\matrix{ {{b_1}} \cr {{b_2}} \cr } } \right]\) be two 2 \(\times\) 1 matrices with real entries such that A = XB, where $$X = {1 \ove...

If the matrices A = \(\left[ {\matrix{ 1 & 1 & 2 \cr 1 & 3 & 4 \cr 1 & { - 1} & 3 \cr } } \right]\),
B = adjA and
C = 3A, then \({{\left| {adjB} \right|} \over {\left| C \right|}}\) is equal to :

If for some \(\alpha\) and \(\beta\) in R, the intersection of the
following three places
x + 4y – 2z = 1
x + 7y – 5z = b
x + 5y + \(\alpha\)z = 5
is a line in R3, then \(\alpha\) + \(\beta\) is equal to :

The following system of linear equations
7x + 6y – 2z = 0
3x + 4y + 2z = 0
x – 2y – 6z = 0, has

The number of all 3 × 3 matrices A, with
enteries from the set {–1, 0, 1} such that the sum
of the diagonal elements of AAT is 3, is

For which of the following ordered pairs (\(\mu\), \(\delta\)),
the system of linear equations
x + 2y + 3z = 1
3x + 4y + 5z = \(\mu\)
4x + 4y + 4z = \(\delta\)
is inconsistent ?

If \(A = \left( {\matrix{ 2 & 2 \cr 9 & 4 \cr } } \right)\) and \(I = \left( {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right)\) then 10A–1 is
equal to :

The system of linear equations
\(\lambda\)x + 2y + 2z = 5
2\(\lambda\)x + 3y + 5z = 8
4x + \(\lambda\)y + 6z = 10 has

If the system of linear equations
2x + 2ay + az = 0
2x + 3by + bz = 0
2x + 4cy + cz = 0,
where a, b, c \(\in\) R are non-zero distinct; has a non-zero solution, then:

Let \(\alpha\) be a root of the equation x2 + x + 1 = 0 and the matrix A = \({1 \over {\sqrt 3 }}\left[ {\matrix{ 1 & 1 & 1 \cr 1 & \alpha & {{\alpha ^2}} \cr 1 & {{\alpha ^2}} & {{\alpha ^4}} \cr } } \right]\) then the ...

Let A = [aij] and B = [bij] be two 3 × 3 real matrices such that bij = (3)(i+j-2)aji, where i, j = 1, 2, 3.
If the determinant of B is 81, then the determinant of A is:

If the system of linear equations,
x + y + z = 6
x + 2y + 3z = 10
3x + 2y + \(\lambda\)z = \(\mu\)
has more than two solutions, then \(\mu\) - \(\lambda\)2
is equal to ______.

The values of \(\lambda\) and \(\mu\) for which the system of linear equations
x + y + z = 2
x + 2y + 3z = 5
x + 3y + \(\lambda\)z = \(\mu\)
has infinitely many solutions are, respectively:

Let m and M be respectively the minimum and maximum values of
$$\left| {\matrix{
{{{\cos }^2}x} & {1 + {{\sin }^2}x} & {\sin 2x} \cr
{1 + {{\cos }^2}x} & {{{\sin }^2}x} & {\sin 2x} \cr
{{{\cos }^2}x} & {{{\sin }^2}x} & {1 + \si...

The sum of distinct values of \(\lambda\) for which the
system of equations\(\left( {\lambda - 1} \right)x + \left( {3\lambda + 1} \right)y + 2\lambda z = 0\)$$\left( {\lambda - 1} \right)x + \left( {4\lambda - 2} \right)y + \left( {\l...

Let \(\theta = {\pi \over 5}\) and \(A = \left[ {\matrix{ {\cos \theta } & {\sin \theta } \cr { - \sin \theta } & {\cos \theta } \cr } } \right]\). If B = A + A4
, then det (B) :

If the minimum and the maximum values of the function \(f:\left[ {{\pi \over 4},{\pi \over 2}} \right] \to R\), defined by
$$f\left( \theta \right) = \left| {\matrix{
{ - {{\sin }^2}\theta } & { - 1 - {{\sin }^2}\theta } & 1 \cr
...

Let \(\lambda \in\) R . The system of linear equations
2x1
- 4x2 + \(\lambda\)x3 = 1
x1 - 6x2 + x3 = 2
\(\lambda\)x1 - 10x2 + 4x3 = 3
is inconsistent for:

If a + x = b + y = c + z + 1, where a, b, c, x, y, z
are non-zero distinct real numbers, then
\(\left| {\matrix{ x & {a + y} & {x + a} \cr y & {b + y} & {y + b} \cr z & {c + y} & {z + c} \cr } } \right|\) is equal to :

If the system of linear equations
x + y + 3z = 0
x + 3y + k2z = 0
3x + y + 3z = 0
has a non-zero solution (x, y, z) for some k \(\in\) R,
then x + \(\left( {{y \over z}} \right)\) is equal to :

If the system of equations
x - 2y + 3z = 9
2x + y + z = b
x - 7y + az = 24, has infinitely many solutions, then a - b is equal to.........

If \(A = \left[ {\matrix{ {\cos \theta } & {i\sin \theta } \cr {i\sin \theta } & {\cos \theta } \cr } } \right]\), \(\left( {\theta = {\pi \over {24}}} \right)\)
and $${A^5} = \left[ {\matrix{
a & b \cr
c & d \cr

}...

Suppose the vectors x1, x2 and x3 are the solutions of the system of linear equations, Ax = b when the vector b on the right side is equal to b1, b2 and b3 respectively. if
$${x_1} = \left[ {\matrix{
1 \cr
1 \cr
1 \cr

} } \...

If the system of equations
x+y+z=2
2x+4y–z=6
3x+2y+\(\lambda\)z=\(\mu\)
has infinitely many solutions, then

Let A = \(\left[ {\matrix{ x & 1 \cr 1 & 0 \cr } } \right]\), x \(\in\) R and A4 = [aij].
If
a11 = 109, then a22 is equal to _______ .

If \(\Delta\) = \(\left| {\matrix{ {x - 2} & {2x - 3} & {3x - 4} \cr {2x - 3} & {3x - 4} & {4x - 5} \cr {3x - 5} & {5x - 8} & {10x - 17} \cr } } \right|\) =
Ax3 + Bx2 + Cx + D, then B + C is equal to :

Let S be the set of all integer solutions, (x, y, z),
of the system of equations
x – 2y + 5z = 0
–2x + 4y + z = 0
–7x + 14y + 9z = 0
such that 15 \(\le\) x2
+ y2
+ z2 \(\le\) 150. Then, the
number of elements in the set S is equal to
...

Let A be a 3 \(\times\) 3 matrix such that
adj A = \(\left[ {\matrix{ 2 & { - 1} & 1 \cr { - 1} & 0 & 2 \cr 1 & { - 2} & { - 1} \cr } } \right]\) and B = adj(adj A).
If |A| = \(\lambda\) and |(B-1)T| = \(\mu\) , then th...

Let S be the set of all \(\lambda\) \(\in\) R for which the system
of linear equations
2x – y + 2z = 2
x – 2y +
\(\lambda\)z = –4
x +
\(\lambda\)y + z = 4
has no solution. Then the set S :

Let A be a 2 \(\times\) 2 real matrix with entries from
{0, 1} and |A|
\(\ne\) 0. Consider the following two
statements :
(P) If A \(\ne\) I2
, then |A| = –1
(Q) If |A| = 1, then tr(A) = 2,
where I2
denotes 2 \(\times\) 2 identity ...

Let A = {X = (x, y, z)T: PX = 0 and
x2 + y2 + z2 = 1} where
\(P = \left[ {\matrix{ 1 & 2 & 1 \cr { - 2} & 3 & { - 4} \cr 1 & 9 & { - 1} \cr } } \right]\),
then the set A :

Let a, b, c \(\in\) R be all non-zero and satisfy
a3 + b3 + c3 = 2. If the matrix
A = \(\left( {\matrix{ a & b & c \cr b & c & a \cr c & a & b \cr } } \right)\)
satisfies ATA = I, then a value of abc can be :

The system of linear equations
x + y + z = 2
2x + 3y + 2z = 5
2x + 3y + (a2 – 1) z = a + 1 then

If \(A = \left[ {\matrix{ {\cos \theta } & { - \sin \theta } \cr {\sin \theta } & {\cos \theta } \cr } } \right]\), then the matrix A–50 when \(\theta\) = \(\pi \over 12\), is equal to :

If   $$A = \left[ {\matrix{
{{e^t}} & {{e^{ - t}}\cos t} & {{e^{ - t}}\sin t} \cr
{{e^t}} & { - {e^{ - t}}\cos t - {e^{ - t}}\sin t} & { - {e^{ - t}}\sin t + {e^{ - t}}co{\mathop{\rm s}\nolimits} t} \cr
{{e^t}} & {2{e^{ - t}}\s...

If the system of linear equations
x \(-\) 4y + 7z = g
       3y \(-\) 5z = h
\(-\)2x + 5y \(-\) 9z = k
is consistent, then :

Let \(\alpha\) and \(\beta\) be the roots of the equation
x2 + x + 1 = 0. Then for y \(\ne\) 0 in R,
$$$\left| {\matrix{
{y + 1} & \alpha & \beta \cr
\alpha & {y + \beta } & 1 \cr
\beta & 1 & {y + \alpha } \cr

} } \...

If \(\left[ {\matrix{ 1 & 1 \cr 0 & 1 \cr } } \right]\left[ {\matrix{ 1 & 2 \cr 0 & 1 \cr } } \right]\)\(\left[ {\matrix{ 1 & 3 \cr 0 & 1 \cr } } \right]\)....$$\left[ {\matrix{
1 & {n - 1} \cr
0 ...

The total number of matrices
\(A = \left( {\matrix{ 0 & {2y} & 1 \cr {2x} & y & { - 1} \cr {2x} & { - y} & 1 \cr } } \right)\)
(x, y \(\in\) R,x \(\ne\) y) for which ATA = 3I3 is :-

If the system of equations 2x + 3y – z = 0, x + ky
– 2z = 0 and 2x – y + z = 0 has a non-trival solution
(x, y, z), then \({x \over y} + {y \over z} + {z \over x} + k\)
is equal to :-