# 15.9: Problems

Problem $$\PageIndex{1}$$

Given

$\mathbf{A}=\begin{pmatrix} 2&3&-1 \\ -5&0&6\\ 0&2&3 \end{pmatrix}\; ;\mathbf{B}=\begin{pmatrix} 2 \\ 1\\0 \end{pmatrix}\; ;\mathbf{C}=\begin{pmatrix} 0&1\\ 2&0\\-1&3 \end{pmatrix} \nonumber$

Multiply all possible pairs of matrices.

Problem $$\PageIndex{2}$$

The matrix representation of a spin $$1/2$$ system was introduced by Pauli in 1926. The Pauli spin matrices are the matrix representation of the angular momentum operator for a single spin $$1/2$$ system and are defined as:

$\mathbf{\sigma_x}=\begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix}\; ;\mathbf{\sigma_y}=\begin{pmatrix} 0&-i \\ i&0 \end{pmatrix}\; ;\mathbf{\sigma_z}=\begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} \nonumber$

1. Show that $$\mathbf{\sigma_x}\mathbf{\sigma_y}=i\mathbf{\sigma_z}$$,$$\mathbf{\sigma_y}\mathbf{\sigma_z}=i\mathbf{\sigma_x}$$ and $$\mathbf{\sigma_z}\mathbf{\sigma_x}=i\mathbf{\sigma_y}$$
2. Calculate the commutator $$\left[\mathbf{\sigma_x},\mathbf{\sigma_y} \right]$$.
3. Show that $$\mathbf{\sigma_x}^2=\mathbf{\sigma_y}^2=\mathbf{\sigma_z}^2=\mathbf{I}$$, where $$\mathbf{I}$$ is the identity matrix. Hint: as with numbers, the square of a matrix is the matrix multiplied by itself.

Problem $$\PageIndex{3}$$

The inversion operator, $$\hat i$$ transforms the point $$(x,y,z)$$ into $$(-x,-y,-z)$$. Write down the matrix that corresponds to this operator.

Problem $$\PageIndex{4}$$

Calculate the inverse of $$\mathbf{A}$$ by definition.

$\mathbf{A}=\begin{pmatrix} 1&-2 \\ 0&1 \end{pmatrix} \nonumber$

Problem $$\PageIndex{5}$$

Calculate the inverse of $$\mathbf{A}$$ by definition.

$\mathbf{A}=\begin{pmatrix} \cos \theta&-\sin \theta \\ \sin \theta&\cos \theta \end{pmatrix} \nonumber$

Problem $$\PageIndex{6}$$

Find the eigenvalues and nomalized eigenvectors of

$\mathbf{M_1}=\begin{pmatrix} 2&0 \\ 0&-3 \end{pmatrix} \nonumber$

$\mathbf{M_2}=\begin{pmatrix} 1&1+i \\ 1-i&1 \end{pmatrix} \nonumber$

Problem $$\PageIndex{7}$$

Given,

$\mathbf{M_3}=\begin{pmatrix} 1&1-i \\ 1+i&1 \end{pmatrix} \nonumber$

1. Show that the matrix is Hermitian.
2. Calculate the eigenvectors and prove they are orthogonal.