A square matrix with non-negative elements and unit row sums is called as a:

Question

Let $ A = \begin{bmatrix} 1 & \sqrt{2} \\ -2 & 0 & 1 \end{bmatrix} $ and $ P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} $, $ \theta > 0 $. If $ B = P A P^T $, $ C = P^T B P $, and the sum of the diagonal elements of $ C $ is $ \frac{m}{n} $, where $ \gcd(m, n) = 1 $, then $ m + n $ is:

Answer 1

Step 1: Test the property on a concrete example. Take $A = \begin{pmatrix} 0.4 & 0.6 \\ 0.7 & 0.3 \end{pmatrix}$. Every entry is non negative and each row adds to 1, so $A$ satisfies the given condition.

Step 2: Check whether $A$ must be unitary (orthogonal, for a real matrix): compute $A A^{T}$. The (1,1) entry is $0.4^2+0.6^2 = 0.16+0.36=0.52 \neq 1$, so $A A^{T} \neq I$. Hence such a matrix need not be unitary, ruling out options (C) and (D).

Step 3: Check positive definiteness: a real positive definite matrix is required to be symmetric, but here $A$ is not symmetric since $0.6 \neq 0.7$. So $A$ cannot be positive definite, ruling out option (B) and (D) again.

Step 4: By elimination, the only property guaranteed for every non negative, unit row sum square matrix is the defining property of a stochastic matrix.

\[\boxed{\text{Stochastic matrix}}\]

A square matrix with non-negative elements and unit row sums is called as a:

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The Correct Option is A

Solution and Explanation

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