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4 problems match “matrix inverse”. Clear

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1. Singular 2x2 matrix parameter ●○○○○ 10 ptsFor which values of $\lambda$ does $A^{-1}$ not exist, where $A=\begin{pmatrix}\lambda&1\\-1&-\lambda\end{pmatrix}$? Give all such values in increasing order.
2. Self-inverse 2x2 matrices ●●○○○ 20 ptsThe matrix $\begin{pmatrix} a & 3 \\ -8 & d \end{pmatrix}$ is its own inverse, for some real numbers $a$ and $d$. Find the number of possible pairs $(a,d)$.
3. Quadruples matching a matrix inverse identity ●●○○○ 20 ptsFind the number of ordered quadruples $(a,b,c,d)$ of real numbers such that $\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \begin{pmatrix} \frac{1}{a} & \frac{1}{b} \\ \frac{1}{c} & \frac{1}{d} \end{pmatrix}$.
4. 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 $a + b + c + d + e + f + g + h$.