Let $A = \begin{bmatrix} 1 & 2 & 1 \\ 0 & 1 & 1\\ 1 & 0 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 1 & 0\\ 1 & 2 & 1\\ 0 & 1 & 1 \end{bmatrix}$. Verify that $AB$ is nonsingular.

Solve the system of equations below using the inverse of a matrix. \begin{align*} x_1+x_2+3x_3+x_4&=5\\ -2x_1-x_2-4x_3-x_4&=-7\\ x_1+4x_2+10x_3+2x_4&=9\\ -2x_1-4x_3+5x_4&=9 \end{align*}

Find values of $x$, $y$, $z$ so that matrix $A = \begin{bmatrix} 1 & 2 & x\\ 3 & 0 & y \\ 1 & 1 & z \end{bmatrix}$ is invertible.

Find values of $x$, $y$ $z$ so that matrix $A = \begin{bmatrix} 1 & x & 1\\ 1 & y & 4 \\ 0 & z & 5 \end{bmatrix}$ is singular.

If $A$ and $B$ are $n \times n$ matrices, $A$ is nonsingular, and $B$ is singular, show directly that $AB$ is singular, without using Theorem NPNT.

Construct an example of a $4\times 4$ unitary matrix.

Matrix multiplication interacts nicely with many operations. But not always with transforming a matrix to reduced row-echelon form. Suppose that $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix. Let $P$ be a matrix that is row-equivalent to $A$ and in reduced row-echelon form, $Q$ be a matrix that is row-equivalent to $B$ and in reduced row-echelon form, and let $R$ be a matrix that is row-equivalent to $AB$ and in reduced row-echelon form. Is $PQ=R$? (In other words, with nonstandard notation, is $\text{rref}(A)\text{rref}(B)=\text{rref}(AB)$?)

Construct a counterexample to show that, in general, this statement is false. Then find a large class of matrices where if $A$ and $B$ are in the class, then the statement is true.

Suppose that $Q$ and $P$ are unitary matrices of size $n$. Prove that $QP$ is a unitary matrix.

Prove that Hermitian matrices (Definition HM) have real entries on the diagonal. More precisely, suppose that $A$ is a Hermitian matrix of size $n$. Then $\matrixentry{A}{ii}\in\real{\null}$, $1\leq i\leq n$.

Suppose that we are checking if a square matrix of size $n$ is unitary. Show that a straightforward application of Theorem CUMOS requires the computation of $n^2$ inner products when the matrix is unitary, and fewer when the matrix is not orthogonal. Then show that this maximum number of inner products can be reduced to $\frac{1}{2}n(n+1)$ in light of Theorem IPAC.

The notation $A^k$ means a repeated matrix product between $k$ copies of the square matrix $A$.

1. Assume $A$ is an $n\times n$ matrix where $A^2=\zeromatrix$ (which does not imply that $A=\zeromatrix$.) Prove that $I_n-A$ is invertible by showing that $I_n+A$ is an inverse of $I_n-A$.
2. Assume that $A$ is an $n\times n$ matrix where $A^3=\zeromatrix$. Prove that $I_n-A$ is invertible.
3. Form a general theorem based on your observations from parts (1) and (2) and provide a proof.