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3 problems match “matrix multiplication”. Clear

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1. Largest sum for an involutory matrix ●●○○○ 20 ptsLet $a,$ $b,$ $c$ be integers such that $\mathbf{A} = \frac{1}{5} \begin{pmatrix} -3 & a \\ b & c \end{pmatrix}$ and $\mathbf{A}^2 = \mathbf{I}.$ Find the largest possible value of $a + b + c.$
2. Quadruples matching a matrix inverse identity ●●○○○ 20 ptsFind the number of ordered quadruples $(a,b,c,d)$ of real numbers such that $\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \begin{pmatrix} \frac{1}{a} & \frac{1}{b} \\ \frac{1}{c} & \frac{1}{d} \end{pmatrix}$.
3. Sum of entries of inverse matrices ●●●○○ 50 ptsThe matrices $\begin{pmatrix} a & 1 & b \\ 2 & 2 & 3 \\ c & 5 & d \end{pmatrix}$ and $\begin{pmatrix} -5 & e & -11 \\ f & -13 & g \\ 2 & h & 4 \end{pmatrix}$ are inverses of each other. Find $a + b + c + d + e + f + g + h$.