Let α be a root of the equation x2 + x + 1 = 0 and the matrix A = 1/√3 [[1, 1, 1], [1, α, α2], [1, α2, α4]], then the matrix A31 is equal to:
Let α, β, γ be the real roots of the equation, x³ + ax² + bx + c = 0, (a, b, c ∈ R and a, b ≠ 0). If the system of equations (in, u, v, w) given by αu + βv + γw = 0, βu + γv + αw = 0; γu + αv + βw = 0 has non-trivial solution, then the value of a²/b is
Consider the following statementsStatement-1 : Given $A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 4 & 1 \\ 2 & 3 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix}$. If $BPA = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$, then $\lambda$ denotes sum of elements of $P$.Statement-2 : Let $\mu$ denote the sum of elements of the matrix $A$ satisfying the matrix equation, $\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} A \begin{bmatrix} 3 & 2 \\ 5 & -3 \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 3 & -1 \end{bmatrix}$Statement-3 : Given that $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 2 & 1 \\ 1 & -1 & 3 \end{bmatrix}$, $C = \begin{bmatrix} 2 & 1 & 1 \\ 2 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$, $D = \begin{bmatrix} 10 \\ 13 \\ 9 \end{bmatrix}$ and that $Cb = D$. If $AX = b$, then $v$ denotes the sum of elements of $X$.Then, which of the following options is/are correct?(A) $2\lambda - 19\mu = 3$(B) $38\mu + 15v = 3$(C) $10v + 8\mu = 2$(D) $\lambda + 19\mu + 38\mu = 0$
Consider a system of linear equations $a_ix + b_iy + c_iz = d_i$ (where $a_i, b_i, c_i \neq 0$ and $i = 1,2,3$ ) & $(\alpha,\beta,\gamma)$ is its unique solution, then match list-I with list-IIList-IList-II(I) If $a_i = k, d_i = k^2, (k \neq 0)$ and $\alpha + \beta + \gamma = 2$, then $k$ is(P) 1(II) If $a_i = d_i = k \neq 0$, then $\alpha + \beta + \gamma$ is(Q) 2(III) If $a_i = k > 0, d_i = k + 1$, then $\alpha + \beta + \gamma$ can be(R) 0(IV) If $a_i = k (S) 3(T) -1