Step 1: Symmetric Matrix Definition. A matrix \( A \) is symmetric if \( A = A^T \). Both matrices \( A \) and \( B \) provided are symmetric.
Step 2: Option Evaluation. - (A) \( A + B \) is symmetric: True. The sum of two symmetric matrices yields a symmetric matrix. - (B) \( AB + BA \) is symmetric: True. The sum \( AB + BA \) is always symmetric. - (C) \( A + A^T \) and \( B + B^T \) are symmetric: True. Given \( A \) and \( B \) are symmetric, \( A + A^T \) and \( B + B^T \) are also symmetric. - (D) \( AB - BA \) is symmetric: False. \( AB - BA \) is typically not symmetric due to the non-commutative nature of matrix multiplication.
Step 3: Final Determination. Statement (D) is incorrect because \( AB - BA \) is generally not a symmetric matrix.
Arrange the following steps in the proper sequence concerning the solution of a linear programming problem.
(A) Graph each constraint as though it were binding, i.e., as if held with strict equality
(B) Find the feasible region, the area of the graph that simultaneously satisfies all constraints
(C) Superimpose contours of the objective function on the feasible region to determine the optimal corner of the region
(D) Construct a graph, placing a decision variable on each axis
Choose the correct answer from the options given below:
Match List-I with List-II
\[\begin{array}{|c|c|}\hline \textbf{List-I} & \textbf{List-II} \\ \hline \text{(A) Closed Interval} & (I)\ [a, b] = \{\,x \in \mathbb{R} : a \leq x \leq b\,\} \\ \hline \text{(B) Open Interval} & (II)\ (a, b) = \{\,x \in \mathbb{R} : a < x < b\,\} \\ \hline \text{(C) Unbounded Interval} & (III)\ [a, b) = \{\,x \in \mathbb{R} : a \leq x < b\,\} \\ \hline \text{(D) Half Open Interval} & (IV)\ (a, \infty) = \{\,x \in \mathbb{R} : a < x\,\} \\ \hline \end{array}\]
Choose the correct answer from the options given below: