If it exists, find the inverse of $A = \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 1 \\ 2 & -1 & 1\end{bmatrix}$, and check your answer.

If it exists, find the inverse of $A = \begin{bmatrix} 2 & -1 & 1\\ 1 & 2 & 1\\3 & 1 & 2\end{bmatrix}$, and check your answer.

If it exists, find the inverse of $A = \begin{bmatrix} 1 & 3 & 1 \\ 1 & 2 & 1 \\ 2 & 2 & 1\end{bmatrix}$, and check your answer.

If it exists, find the inverse of $A = \begin{bmatrix} 1 & 3 & 1 \\ 0 & 2 & 1 \\ 2 & 2 & 1\end{bmatrix}$, and check your answer.

Verify that $B$ is the inverse of $A$. \begin{align*} A&= \begin{bmatrix} 1 & 1 & -1 & 2\\ -2 & -1 & 2 & -3\\ 1 & 1 & 0 & 2\\ -1 & 2 & 0 & 2 \end{bmatrix} & B&= \begin{bmatrix} 4 & 2 & 0 & -1\\ 8 & 4 & -1 & -1\\ -1 & 0 & 1 & 0\\ -6 & -3 & 1 & 1 \end{bmatrix} \end{align*}

Recycle the matrices $A$ and $B$ from Exercise MISLE.C21 and set \begin{align*} \vect{c}&=\colvector{2\\1\\-3\\2}&\vect{d}&=\colvector{1\\1\\1\\1} \end{align*} Employ the matrix $B$ to solve the two linear systems $\linearsystem{A}{\vect{c}}$ and $\linearsystem{A}{\vect{d}}$.

If it exists, find the inverse of the $2\times 2$ matrix \begin{align*} A=\begin{bmatrix} 7&3\\5&2 \end{bmatrix} \end{align*} and check your answer. (See Theorem TTMI.)

If it exists, find the inverse of the $2\times 2$ matrix \begin{align*} A=\begin{bmatrix} 6&3\\4&2 \end{bmatrix} \end{align*} and check your answer. (See Theorem TTMI.)

At the conclusion of Example CMI, verify that $BA=I_5$ by computing the matrix product.

Let \begin{equation*} D=\begin{bmatrix} 1 & -1 & 3 & -2 & 1\\ -2 & 3 & -5 & 3 & 0\\ 1 & -1 & 4 & -2 & 2\\ -1 & 4 & -1 & 0 & 4\\ 1 & 0 & 5 & -2 & 5 \end{bmatrix} \end{equation*} Compute the inverse of $D$, $\inverse{D}$, by forming the $5\times 10$ matrix $\augmented{D}{I_5}$ and row-reducing (Theorem CINM). Then use a calculator to compute $\inverse{D}$ directly.

Let \begin{equation*} E=\begin{bmatrix} 1 & -1 & 3 & -2 & 1\\ -2 & 3 & -5 & 3 & -1\\ 1 & -1 & 4 & -2 & 2\\ -1 & 4 & -1 & 0 & 2\\ 1 & 0 & 5 & -2 & 4 \end{bmatrix} \end{equation*} Compute the inverse of $E$, $\inverse{E}$, by forming the $5\times 10$ matrix $\augmented{E}{I_5}$ and row-reducing (Theorem CINM). Then use a calculator to compute $\inverse{E}$ directly.

Let \begin{equation*} C= \begin{bmatrix} 1 & 1 & 3 & 1\\ -2 & -1 & -4 & -1\\ 1 & 4 & 10 & 2\\ -2 & 0 & -4 & 5 \end{bmatrix} \end{equation*} Compute the inverse of $C$, $\inverse{C}$, by forming the $4\times 8$ matrix $\augmented{C}{I_4}$ and row-reducing (Theorem CINM). Then use a calculator to compute $\inverse{C}$ directly.

Find all solutions to the system of equations below, making use of the matrix inverse found in Exercise MISLE.C28. \begin{align*} x_1+x_2+3x_3+x_4&=-4\\ -2x_1-x_2-4x_3-x_4&=4\\ x_1+4x_2+10x_3+2x_4&=-20\\ -2x_1-4x_3+5x_4&=9 \end{align*}

Use the inverse of a matrix to find all the solutions to the following system of equations. \begin{align*} x_1 + 2 x_2 - x_3 &= -3\\ 2 x_1 + 5 x_2 - x_3 &= -4\\ -x_1 - 4 x_2 &= 2 \end{align*}

Use a matrix inverse to solve the linear system of equations. \begin{align*} x_1-x_2+2x_3&=5\\ x_1-2x_3&=-8\\ 2x_1-x_2-x_3&=-6 \end{align*}

Construct an example to demonstrate that $\inverse{(A+B)}=\inverse{A}+\inverse{B}$ is not true for all square matrices $A$ and $B$ of the same size.