If for the matrix, A = \left[ {\matrix{ 1 & { - \alpha } \cr \alpha & \beta \cr } } \right], $$A{A}^T=I_2$$, then the value of $${\alpha }^4+{\beta }^4$$ is :

The following system of linear equations 2x + 3y + 2z = 9 3x + 2y + 2z = 9 x $$-$$ y + 4z = 8

Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A2 is 1, then the possible number of such matrices is :

The value of \left| {\matrix{ {(a + 1)(a + 2)} & {a + 2} & 1 \cr {(a + 2)(a + 3)} & {a + 3} & 1 \cr {(a + 3)(a + 4)} & {a + 4} & 1 \cr } } \right| is :

Consider the following system of equations : x + 2y $$-$$ 3z = a 2x + 6y $$-$$ 11z = b x $$-$$ 2y + 7z = c, where a, b and c are real constants. Then the system of equations :

Let A = \left[ {\matrix{ i & { - i} \cr { - i} & i \cr } } \right],i = \sqrt { - 1}. Then, the system of linear equations $$A^8\left[\text{matrix}x\text{cr}y\text{cr}\right]=\left[\text{matrix}8\text{cr}64\text{cr}\right]$$ has :

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 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 the value of k2 is :

The solutions of the equation \left| {\matrix{ {1 + {{\sin }^2}x} & {{{\sin }^2}x} & {{{\sin }^2}x} \cr {{{\cos }^2}x} & {1 + {{\cos }^2}x} & {{{\cos }^2}x} \cr {4\sin 2x} & {4\sin 2x} & {1 + 4\sin 2x} \cr } } \right| = 0,(0 < x < \pi ), are

Let $$\alpha$$, $$\beta$$, $$\gamma$$ be the real roots of the equation, x3 + ax2 + bx + c = 0, (a, b, c $$\in$$ R and a, b $$\ne$$ 0). If the system of equations (in u, v, w) given by $$\alpha$$u + $$\beta$$v + $$\gamma$$w = 0, $$\beta$$u + $$\gamma$$v + $$\alpha$$w = 0; $$\gamma$$u + $$\alpha$$v + $$\beta$$w = 0 has non-trivial solution, then the value of {{{a^2}} \over b} is

Let A + 2B = \left[ {\matrix{ 1 & 2 & 0 \cr 6 & { - 3} & 3 \cr { - 5} & 3 & 1 \cr } } \right] and 2A - B = \left[ {\matrix{ 2 & { - 1} & 5 \cr 2 & { - 1} & 6 \cr 0 & 1 & 2 \cr } } \right]. If Tr(A) denotes the sum of all diagonal elements of the matrix A, then Tr(A) $$-$$ Tr(B) has value equal to

Let the system of linear equations 4x + $$\lambda$$y + 2z = 0 2x $$-$$ y + z = 0 $$\mu$$x + 2y + 3z = 0, $$\lambda$$, $$\mu$$$$\in$$R. has a non-trivial solution. Then which of the following is true?

Let A = \left[ {\matrix{ 2 & 3 \cr a & 0 \cr } } \right], a$$\in$$R be written as P + Q where P is a symmetric matrix and Q is skew symmetric matrix. If det(Q) = 9, then the modulus of the sum of all possible values of determinant of P is equal to :

The value of k $$\in$$R, for which the following system of linear equations 3x $$-$$ y + 4z = 3, x + 2y $$-$$ 3z = $$-$$2 6x + 5y + kz = $$-$$3, has infinitely many solutions, is :

The values of $$\lambda$$ and $$\mu$$ such that the system of equations $$x+y+z=6$$, $$3x+5y+5z=26$$, $$x+2y+\lambda z=\mu$$ has no solution, are :

Let A = [aij] be a real matrix of order 3 $$\times$$ 3, such that ai1 + ai2 + ai3 = 1, for i = 1, 2, 3. Then, the sum of all the entries of the matrix A3 is equal to :

The values of a and b, for which the system of equations 2x + 3y + 6z = 8 x + 2y + az = 5 3x + 5y + 9z = b has no solution, are :

Let A and B be two 3 $$\times$$ 3 real matrices such that (A2 $$-$$ B2) is invertible matrix. If A5 = B5 and A3B2 = A2B3, then the value of the determinant of the matrix A3 + B3 is equal to :

The number of distinct real roots of \left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right| = 0 in the interval - {\pi \over 4} \le x \le {\pi \over 4} is :