Let k be a positive real number and let A \textrm{ }\textrm{ } = \textrm{ }\textrm{ } \left[\right. 2 k - 1 \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } 2 \sqrt{k} \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } 2 \sqrt{k} \\ 2 \sqrt{k} \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } 1 \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } - 2 k \\ - 2 \sqrt{k} \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } 2 k \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } - 1 \left]\right. \textrm{ }\textrm{ }\textrm{ } and \textrm{ }\textrm{ }\textrm{ } B \textrm{ }\textrm{ } = \textrm{ }\textrm{ } \left[\right. \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } 0 \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } 2 k - 1 \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } \sqrt{k} \\ 1 - 2 k \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } 0 \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } 2 \sqrt{k} \\ \textrm{ }\textrm{ } - \sqrt{k} \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } - 2 \sqrt{k} \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } 0 \left]\right. If det (adj A) + det(adj B) =106, 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].

Engineering

Mathematics

Transpose and Adjoint of a Matrix

Question

Let k be a positive real number and let

$A = [\begin{array}{l} 2 k - 1 2 \sqrt{k} 2 \sqrt{k} \\ 2 \sqrt{k} 1 - 2 k \\ - 2 \sqrt{k} 2 k - 1 \end{array}] and B = [\begin{array}{l} 0 2 k - 1 \sqrt{k} \\ 1 - 2 k 0 2 \sqrt{k} \\ - \sqrt{k} - 2 \sqrt{k} 0 \end{array}]$

If det (adj A) + det(adj B) =10⁶, 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].

LiveFree JEE Main Previous Year Online Papers Live Free JEE Advance Previous Year Online Papers

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

R₂ $\to_{}^{}$ R₂ + R₃

_{$| A | = | \begin{matrix} 2 k - 1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 0 & 1 + 2 k & - (2 k + 1) \\ - 2 \sqrt{k} & 2 k & - 1 \end{matrix} |$}

C₂ $\to_{}^{}$ C₂ + C₃

_{$= | \begin{matrix} 2 k - 1 & 4 \sqrt{k} & 2 \sqrt{k} \\ 0 & 0 & - (2 k + 1) \\ - 2 \sqrt{k} & 2 k - 1 & - 1 \end{matrix} | = (2 k + 1) {{(2 k - 1)}^{2} + 8 k} = (2 k + 1) {4 k^{2} - 4 k + 1} = 8 k$}

= (2k + 1)³

B| = 0

det(adj A) + det (adj B) = 10⁶

^{${({(2 k + 1)}^{3})}^{3 - 1} + 0 = 10^{6} \Rightarrow {(2 k + 1)}^{6} = 10^{6} \Rightarrow 2 k + 1 = 10$}

^{$\Rightarrow k = \frac{9}{2} ∴ [x] = 4$}