In the following, $A$, $B$, $C$, $\ldots$ are matrices of appropriate sizes, depending on the context.

  1. Prove the following statements:
    1. Prove that if $AA^\prime=I$ and $A^{\prime\prime}A=I$, then $A^\prime=A^{\prime\prime}$. (The left inverse equals the right inverse)
    2. Prove that if $A$ is invertible, then $AB=O$ implies $B=O$ where $O$ iz the zero matrix.
    3. Show that $(AB)^{-1}=B^{-1}A^{-1}$ if $A$ and $B$ are invertible. (Relationship between multiplication and inversion)
  2. Prove that $(AB)^\intercal=B^\intercal A^\intercal$ for properly defined matrices $A$ and $B$.
  3. Prove $A(B+C)=AB+AC$
  4. Let $A=\begin{bmatrix}a & b\\c & d\end{bmatrix}$.
    1. Find the inverse of $A$ if it exists.
    2. Show that if $Ax=b$ has a unique solution if and only if $\det(A)\neq 0$.
  5. By trial and error find examples of $2$ by $2$ matrices such that
    1. $A^2=-I$, $A$ having only real entries.
    2. $B^2=0$, although $B\neq 0$.
    3. $CD=-DC$, not allowing the case $CD=0$.
    4. $EF=0$, although no entries of $E$ or $F$ are zero.

  6. The matrix that rotates the $xy-$plane by an angle $\theta$ is

    \[A(\theta)=\begin{bmatrix} \cos(\theta) & -\sin(\theta)\\\\ \sin(\theta) & \cos(\theta) \end{bmatrix}.\]

    Show that $A(\theta_1)A(\theta_2)=A(\theta_1+\theta_2)$.

  7. Which three matrix $E_1$, $E_2$, and $E_3$ put $A$ into a upper triangular matrix $U?$

    \[A=\begin{bmatrix} 1 & 1 & 0\\\\ 4& 6 & 1\\\\ -2 & 2 & 0 \end{bmatrix}\quad\text{and}\quad E_3E_2E_1A=U\]
  8. The parabola $y=a+bx+cx^2$ goes through the point $(x,y)=(1,4)$ and $(2,8)$ and $(3,14)$. Find and solve a matrix equation for the unknowns $(a,b,c).$