# Questions 11 - 16#

## Q11#

Find $$\pmb{AB}$$ and $$\pmb{BA}$$ if $$\displaystyle \pmb{A} =\begin{bmatrix} 1 & 2\\ 3& 4\end{bmatrix}$$ and $$\displaystyle \pmb{B} =\begin{bmatrix} 5 & 6\\ 7& 8\end{bmatrix}$$ . Do these matrices commute and if not what is $$[\pmb{AB}]$$?

## Q12#

(a) Find $$\pmb{A}^2,\pmb{B}^2,\pmb{AB},\pmb{BA},\pmb{A}^2\pmb{B}$$,and $$\pmb{B}^2\pmb{A}$$ if $$\displaystyle \pmb{A} =\begin{bmatrix} 1 & 1\\ 0& 1\end{bmatrix}$$ and $$\displaystyle \pmb{B} =\begin{bmatrix} 1 & 0\\ 1& 1\end{bmatrix}$$.

(b) Do $$\pmb{A}^2\pmb{B}$$ and $$\pmb{B}^2\pmb{A}$$ commute ?

## Q13#

If $$\pmb{x}=[x_1, x_2 ],\pmb{y}=[y_1,y_2]^T$$ ,and $$\displaystyle \pmb{Q}=\begin{bmatrix} 1 &-4\\ -9& 6\end{bmatrix}$$, find $$\pmb{xQy}$$ and $$\pmb{yQx}$$.

## Q14#

(a) Explain why if $$j$$ is a constant factor that $$|j\pmb{M}| = j^n|\pmb{M}|$$ for an $$n \times n$$ matrix.

(b) Confirm equation 4 if $$\displaystyle \pmb{A} =\begin{bmatrix} a & b\\ c& d\end{bmatrix}$$ by calculating $$\pmb{A}^{-1}$$ and $$| \pmb{A}^{-1} |$$.

(c) Using Sympy (or by hand ) find $$\pmb{A}^{-1}$$ and $$|\pmb{A}^{-1} |$$ for the $$3\times 3$$ matrix $$\displaystyle \pmb{A} =\begin{bmatrix} 0 & b & c\\ b& 0 & d\\ c & d & 0\end{bmatrix}$$.

Strategy: See Section 4.14, but now the problem is algebraic not numerical.

## Q15 Commuting operators#

(a) Show that if $$\pmb{P}$$ and $$\pmb{Q}$$ are linear operators, not necessarily matrices, $$[\pmb{P},\pmb{Q}]=- [\pmb{Q},\pmb{P}]$$.

(b) (i) find $$[ d/dx, x]\sin(x)$$ , (ii) $$[d/dx, x]f(x)$$, and (iii) $$[d/dx, x]$$ and

(c) Show that operators $$d/dx$$ and $$x$$ do not commute.

(d) Write a Python/Sympy commutator function and test your results.

(e) Do $$\displaystyle \frac{df(x)}{dx}$$ and $$\displaystyle \int_a^bf(x)dx$$ commute?

(f) Does the operator $$df/dx$$ and a displacement operator $$\Delta f = f (x + c)$$ commute?

(g) Does the operator $$xx$$ which means multiply twice by $$x$$, commute with the inversion operator, $$Inv( f(x)) = f (-x)$$?

## Q16 Commuting matrices#

$$\pmb{B}$$ and $$\pmb{C}$$ are commuting square matrices and the matrix $$\pmb{A}$$ is defined as $$\pmb{A} \equiv e^{\pmb{B}} =\pmb{1}+\pmb{B}+\pmb{B}^2/2! +\cdots$$ show that:

(a) $$e^{\pmb{B}}e^{\pmb{C}} = e^{\pmb{B+C}}$$,

(b) $$\pmb{A}^{-1} = e^{-\pmb{B}}$$,

(c) $$e^{\pmb{CBC}^{-1}} = \pmb{CAC}^{-1}$$.

Strategy: (a) Expand out the exponentials as shown in Section 5.5 and collect terms and try to reform an exponential series in $$\pmb{B + C}$$. (b) Use the fact that for an inverse matrix $$\pmb{AA}^{-1} = \pmb{1}$$. (c) The expression $$\pmb{CAC}^{-1}$$ is a similarity transform and is itself a square matrix. To prove this result, expand the exponential on both sides of the equation