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Inverses and singular matrices

Solve the first problem 17 problems

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1. Value of x making a matrix singular ●○○○○ 10 ptsFind the value of $x$ for which the matrix $\begin{pmatrix} 1 + x & 7 \\ 3 - x & 8 \end{pmatrix}$ is not invertible. Round to two decimal places.
2. 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.
3. 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 $a + b + c.$
4. 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 $(a,d)$.
5. Quadruples matching a matrix inverse identity ●●○○○ 20 ptsFind the number of ordered quadruples $(a,b,c,d)$ of real numbers such that $\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \begin{pmatrix} \frac{1}{a} & \frac{1}{b} \\ \frac{1}{c} & \frac{1}{d} \end{pmatrix}$.
6. Smallest entry sum for a conjugating matrix ●●●○○ 50 ptsFor some matrix $\mathbf{P} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with nonzero integer entries, $\begin{pmatrix} 4 & 3 \\ -1 & 0 \end{pmatrix} = \mathbf{P}^{-1} \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix} \mathbf{P}$. Find the smallest possible value of $|a| + |b| + |c| + |d|$.
7. 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 $a + b + c + d + e + f + g + h$.
8. Scaling matrix powers to converge ●●●○○ 50 ptsLet $A=\begin{pmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{pmatrix}$. Find the real number $a$ for which the limit $\lim_{n\to\infty} a^nA^n$ exists and is non-zero, where a sequence of matrices $B_n$ converges to $B$ when $B_nx\to Bx$ for every $x\in\mathbb{R}^3$. Give $a$ rounded to two decimal places.

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