Matrix inverses

For each of the following pairs of matrices, determine whether $B$ is an inverse of $A$.

Suppose $AB=AC$ and $A$ is an invertible $n\times n$-matrix. Does it follow that $B=C$? Explain why or why not.

Suppose $AB=AC$ and $A$ is a non-invertible $n\times n$-matrix. Does it follow that $B=C$? Explain why or why not.

For each of the following matrices, find the inverse if possible. If the inverse does not exist, explain why.

For each of the following matrices, find the inverse if possible. If the inverse does not exist, explain why.

Let $A$ be a $2\times 2$ invertible matrix, with $A=\left[\begin{array}{cc}a&b\\ c&d\end{array}\right]$. Find a formula for $A^{-1}$ in terms of $a,b,c,d$.

Using the inverse of the matrix, find the solution to the systems:

Now give the solution in terms of $a,b$, and $c$ to the following:

Show that if $A$ is an $n\times n$ invertible matrix and $X$ and $B$ are $n\times 1$-matrices such that $AX=B$, then $X=A^{-1}B$.

Prove that if $A^{-1}$ exists and $AX=0$ then $X=0$.

Show $(AB)^{-1}=B^{-1}A^{-1}$ by verifying that

Is it possible to have matrices $A$ and $B$ such that $AB=I$, while $BA=0$? If it is possible, give an example of such matrices. If it is not possible, explain why.

If $A$ is invertible, show that $A^{2}$ is invertible and $(A^{2})^{-1}=(A^{-1})^{2}$.

If $A$ is invertible, show $(A^{-1})^{-1}=A$. Hint: Use the uniqueness of inverses.

Determine whether $B$ is a right inverse, left inverse, both, or neither of $A$.

Show that right inverses are not unique by giving an example of matrices $A,B,C$ such that both $B$ and $C$ are right inverses of $A$, but $B\neq C$.

Solve the following system of equations by using the inverse of a suitable matrix.

Suppose that $A,B,C,D$ are $n\times n$-matrices, and that all relevant matrices are invertible. Further, suppose that $(A+B)^{-1}=CB^{-1}$. Solve this equation for $A$ (in terms of $B$ and $C$), $B$ (in terms of $A$ and $C$), and $C$ (in terms of $A$ and $B$).

Which of the following matrices is right invertible? Find a right inverse if one exists. If possible, find two different right inverses.