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) $\lambda + 19\mu + 38\mu = 0$
Which of the following options is/are correct?(A) Let $A = \begin{bmatrix} 2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -x & 14x & 7x \\ 0 & 1 & 0 \\ x & -4x & -2x \end{bmatrix}$ are two matrices such that $AB = (AB)^{-1}$ and $AB \neq I$ (where $I$ is an identity matrix of order $3 \times 3$). Then $tr\left(AB + (AB)^2 + (AB)^3 + ..... + (AB)^{100}\right)$ equals 100.(B) If $A$ and $B$ are square matrices of order 3, where $|A| = -2$ and $|B| = 1$, then $|(A^{-1})adj(B^{-1})adj(2A^{-1})|$ equals 8.(C) If $F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then $F(x).F(y) = F(x + y)$ and $[F(x)]^{-1} = F(-x)$.(D) Let $X$ be the solution set of the equation $A^x = I$, where $A = \begin{bmatrix} 0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{bmatrix}$ and $I$ is the corresponding unit matrix and $X \subseteq \mathbb{N}$ then the minimum value of $\sum(\cos^n \theta + \sin^n \theta), \theta \in \mathbb{R}, n \in \mathbb{Z}$ is 2.
Let A be a 2 x 2 matrix with non-zero entries and let A2 = I, where I is 2 x 2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A| = determinant of matrix A.Statement-1 : Tr(A) = 0.Statement-2 : |A| = 1.(A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for statement-1.(B) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for statement-1.(C) Statement-1 is true, Statement-2 is false.(D) Statement-1 is false, Statement-2 is true.
Let a, b, c, l, m, n ∈ R such that al + bm + cn = 0, bl + cm + an = 0, cl + am + bn = 0. If a, b & c are distinct & f(x) = ax^3 + bx^2 + cx + 5, then the value of f(1) is
Let a1, a2, a3, ..., a10 be in G.P. with ai > 0 for i = 1,2,..., 10 and S be the set of pairs (r, k), r, k ∈ N (the set of natural numbers) for which . Then the number of elements in S, is :