If \(B\) is a non-singular \(4 \times 4\) matrix and \(A\) is its adjoint such that \(|A| = 125\), then \(|B|\) is:

Question

If \[ A = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \] and \(\theta = \frac{2\pi}{7}\), then \(A^{100} = A \times A \times \ldots\) (100 times) is equal to:

Answer 1

The relationship between a square matrix \(B\), its adjoint \(A\), and their determinants is expressed as:\[|A| = |\text{adj} B| = |B|^{n-1}\]Here, \(n\) denotes the order of the matrix, with \(n=4\).We are given:\[|A| = 125, \quad n=4\]Substituting these values into the equation:\[125 = |B|^{4-1} = |B|^3\]To find \(|B|\), we take the cube root of both sides:\[|B| = \sqrt[3]{125} = 5\]

If \(B\) is a non-singular \(4 \times 4\) matrix and \(A\) is its adjoint such that \(|A| = 125\), then \(|B|\) is:

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The Correct Option is A

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