# Paper 1, Section II, $8 \mathbf{C}$

(a) Show that the equations

$\begin{array}{r} 1+s+t=a \\ 1-s+t=b \\ 1-2 t=c \end{array}$

determine $s$ and $t$ uniquely if and only if $a+b+c=3$.

Write the following system of equations

\begin{aligned} &5 x+2 y-z=1+s+t \\ &2 x+5 y-z=1-s+t \\ &-x-y+8 z=1-2 t \end{aligned}

in matrix form $A \mathbf{x}=\mathbf{b}$. Use Gaussian elimination to solve the system for $x, y$, and $z$. State a relationship between the rank and the kernel of a matrix. What is the rank and what is the kernel of $A$ ?

For which values of $x, y$, and $z$ is it possible to solve the above system for $s$ and $t$ ?

(b) Define a unitary $n \times n$ matrix. Let $A$ be a real symmetric $n \times n$ matrix, and let $I$ be the $n \times n$ identity matrix. Show that $|(A+i I) \mathbf{x}|^{2}=|A \mathbf{x}|^{2}+|\mathbf{x}|^{2}$ for arbitrary $\mathbf{x} \in \mathbb{C}^{n}$, where $|\mathbf{x}|^{2}=\sum_{j=1}^{n}\left|x_{j}\right|^{2}$. Find a similar expression for $|(A-i I) \mathbf{x}|^{2}$. Prove that $(A-i I)(A+i I)^{-1}$ is well-defined and is a unitary matrix.