AP DSC School Assistant Math Model Test 2

Concept:The main idea is to test properties of matrices: triangular matrices and their inverses, adjoint properties, and product of adjoints.Explanation:We examine each option step by step.Option (A): A triangular matrix (upper or lower) remains triangular after inversion. For an upper triangular matrix $A = \begin{bmatrix} a & b \\ 0 & c \end{bmatrix}$, its inverse is $A^{-1} = \frac{1}{ac} \begin{bmatrix} c & -b \\ 0 & a \end{bmatrix}$, which is also upper triangular. This holds for any size. Hence statement (A) is correct.Option (B): For any matrix $A$, $\operatorname{adj}(A^T) = (\operatorname{adj} A)^T$, not $\operatorname{adj} A$ in general. Taking transpose of a matrix does not leave the adjoint unchanged unless $A$ is symmetric. So statement (B) is false.Option (C): For square matrices $A$ and $B$ of order $n$, $\operatorname{adj}(AB) = (\operatorname{adj} B)(\operatorname{adj} A)$ (the order is reversed), not $(\operatorname{adj} A)(\operatorname{adj} B)$. A simple counterexample with $2\times2$ matrices shows the given equality fails. Hence (C) is false.Answer:Option A is correct.