Then which of the following statements is (are) TRUE?

Let $\mathbb{R}^2$ denote $\mathbb{R} \times \mathbb{R}$. Let

$$ S=\left\{(a, b, c): a, b, c \in \mathbb{R} \text { and } a x^2+2 b x y+c y^2>0 \text { for all }(x, y) \in \mathbb{R}^2-\{(0,0)\}\right\} . $$

Then which of the following statements is (are) TRUE?

Let a, $$\lambda$$, m $$\in$$ R. Consider the system of linear equations

ax + 2y = $$\lambda$$

3x $$-$$ 2y = $$\mu$$

Which of the following statements is(are) correct?

Let $$P = \left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} & 0 \cr } } \right]$$, where $$\alpha$$ $$\in$$ R. Suppose $$Q = [{q_{ij}}]$$ is a matrix such that PQ = kl, where k $$\in$$ R, k $$\ne$$ 0 and I is the identity matrix of order 3. If $${q_{23}} = - {k \over 8}$$ and $$\det (Q) = {{{k^2}} \over 2}$$, then

Let X and Y be two arbitrary, 3 $$\times$$ 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 $$\times$$ 3, non-zero, symmetric matrix. Then which of the following matrices is(are) skew symmetric?

Which of the following values of $$\alpha$$ satisfy the equation

$$\left| {\matrix{ {{{(1 - \alpha )}^2}} & {{{(1 + 2\alpha )}^2}} & {{{(1 + 3\alpha )}^2}} \cr {{{(2 + \alpha )}^2}} & {{{(2 + 2\alpha )}^2}} & {{{(2 + 3\alpha )}^2}} \cr {{{(3 + \alpha )}^2}} & {{{(3 + 2\alpha )}^2}} & {{{(3 + 3\alpha )}^2}} \cr } } \right| = - 648\alpha $$ ?

Let $$\omega$$ be a complex cube root of unity with $$\omega$$ $$\ne$$ 1 and P = [p_ij] be a n $$\times$$ n matrix with p_ij = $$\omega$$^{i + j}. Then P² $$\ne$$ 0, when n = ?

If the ad joint of a 3 $$\times$$ 3 matrix P is $$\left[ {\matrix{ 1 & 4 & 4 \cr 2 & 1 & 7 \cr 1 & 1 & 3 \cr } } \right]$$, then the possible value(s) of the determinant of P is(are)

Let M and N be two 3 $$\times$$ 3 non-singular skew symmetric matrices such that MN = NM. If P^T denotes the transpose of P, then M²N²(M^TN)^$$-$$1(MN^$$-$$1)^T is equal to

Consider the matrix

$$ P = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}. $$

Let the transpose of a matrix $X$ be denoted by $X^T$. Then the number of $3 \times 3$ invertible matrices $Q$ with integer entries, such that

$$ Q^{-1} = Q^T \quad \text{and} \quad PQ = QP, $$

is

Let $$p, q, r$$ be nonzero real numbers that are, respectively, the $$10^{\text {th }}, 100^{\text {th }}$$ and $$1000^{\text {th }}$$ terms of a harmonic progression. Consider the system of linear equations

$$$ \begin{gathered} x+y+z=1 \\ 10 x+100 y+1000 z=0 \\ q r x+p r y+p q z=0 \end{gathered} $$$

The correct option is:

Let $$P = \left[ {\matrix{ 1 & 0 & 0 \cr 4 & 1 & 0 \cr {16} & 4 & 1 \cr } } \right]$$ and I be the identity matrix of order 3. If $$Q = [{q_{ij}}]$$ is a matrix such that $${P^{50}} - Q = I$$ and $${{{q_{31}} + {q_{32}}} \over {{q_{21}}}}$$ equals

If P is a 3 $$\times$$ 3 matrix such that P^T = 2P + I, where P^T is the transpose of P and I is the 3 $$\times$$ 3 identity matrix, then there exists a column matrix $$X = \left[ {\matrix{ x \cr y \cr z \cr } } \right] \ne \left[ {\matrix{ 0 \cr 0 \cr 0 \cr } } \right]$$ such that

Let $$P = [{a_{ij}}]$$ be a 3 $$\times$$ 3 matrix and let $$Q = [{b_{ij}}]$$, where $${b_{ij}} = {2^{i + j}}{a_{ij}}$$ for $$1 \le i,j \le 3$$. If the determinant of P is 2, then the determinant of the matrix Q is

If the point P(a, b, c), with reference to (E), lies on the plane 2x + y + z = 1, then the value of 7a + b + c is

Let $$\omega$$ be a solution of $${x^3} - 1 = 0$$ with $${\mathop{\rm Im}\nolimits} (\omega ) > 0$$. If a = 2 with b and c satisfying (E), then the value of $${3 \over {{\omega ^a}}} + {1 \over {{\omega ^b}}} + {3 \over {{\omega ^c}}}$$ is equal to

Let b = 6, with a and c satisfying (E). If $$\alpha$$ and $$\beta$$ are the roots of the quadratic equation ax² + bx + c = 0, then $$\sum\limits_{n = 0}^\infty {{{\left( {{1 \over \alpha } + {1 \over \beta }} \right)}^n}} $$ is

Let $$\omega$$ $$\ne$$ 1 be a cube root of unity and S be the set of all non-singular matrices of the form $$\left[ {\matrix{ 1 & a & b \cr \omega & 1 & c \cr {{\omega ^2}} & \omega & 1 \cr } } \right]$$, where each of a, b, and c is either $$\omega$$ or $$\omega$$². Then the number of distinct matrices in the set S is

The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system

$\mathrm{A}\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ has exactly two distinct solutions, is

The number of A in $\mathrm{T}_p$ such that the trace of A is not divisible by $p$ but $\operatorname{det}(\mathrm{A})$ is divisible by $p$ is

[Note : The trace of a matrix is the sum of its diagonal entries.]

The number of matrices in A is

The number of matrices A in A for which the system of linear equations $$A\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]$$ has a unique solution, is

The number of matrices A in A for which the system of linear equations $$A\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]$$ is inconsistent, is

Consider the system of equations:

$$x-2y+3z=-1$$

$$-x+y-2z=k$$

$$x-3y+4z=1$$

Statement - 1 : The system of equations has no solution for $$k\ne3$$.

and

Statement - 2 : The determinant $$\left| {\matrix{ 1 & 3 & { - 1} \cr { - 1} & { - 2} & k \cr 1 & 4 & 1 \cr } } \right| \ne 0$$, for $$k \ne 3$$.

Let $S=\left\{A=\left(\begin{array}{lll}0 & 1 & c \\ 1 & a & d \\ 1 & b & e\end{array}\right): a, b, c, d, e \in\{0,1\}\right.$ and $\left.|A| \in\{-1,1\}\right\}$, where $|A|$ denotes the determinant of $A$. Then the number of elements in $S$ is __________.

The total number of distinct x $$\in$$ R for which

$$\left| {\matrix{ x & {{x^2}} & {1 + {x^3}} \cr {2x} & {4{x^2}} & {1 + 8{x^3}} \cr {3x} & {9{x^2}} & {1 + 27{x^3}} \cr } } \right| = 10$$ is ______________.

Let $$z = {{ - 1 + \sqrt 3 i} \over 2}$$, where $$i = \sqrt { - 1} $$, and r, s $$\in$$ {1, 2, 3}. Let $$P = \left[ {\matrix{ {{{( - z)}^r}} & {{z^{2s}}} \cr {{z^{2s}}} & {{z^r}} \cr } } \right]$$ and I be the identity matrix of order 2. Then the total number of ordered pairs (r, s) for which P² = $$-$$I is ____________.

Let M be a 3 $$\times$$ 3 matrix satisfying $$M\left[ {\matrix{ 0 \cr 1 \cr 0 \cr } } \right] = \left[ {\matrix{ { - 1} \cr 2 \cr 3 \cr } } \right]$$, $$M\left[ {\matrix{ 1 \cr { - 1} \cr 0 \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr { - 1} \cr } } \right]$$ and $$M\left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right] = \left[ {\matrix{ 0 \cr 0 \cr {12} \cr } } \right]$$. Then the sum of the diagonal entries of M is ___________.

Let $k$ be a positive real number and let

$$ \begin{aligned} A & =\left[\begin{array}{ccc} 2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & -2 k \\ -2 \sqrt{k} & 2 k & -1 \end{array}\right] \text { and } \\\\ \mathbf{B} & =\left[\begin{array}{ccc} 0 & 2 k-1 & \sqrt{k} \\ 1-2 k & 0 & 2 \sqrt{k} \\ -\sqrt{k} & -2 \sqrt{k} & 0 \end{array}\right] . \end{aligned} $$

If $\operatorname{det}(\operatorname{adj} A)+\operatorname{det}(\operatorname{adj} B)=10^6$, then $[k]$

is equal to _________.

[ Note : adj M denotes the adjoint of a square matrix M and $[k]$ denotes the largest integer less than or equal to $k$ ].