Supplementary Problems

Let $A$, $B$ be $2 \times 2$ matrices over $\mathbb {C}$. What are the conditions on $A$, $B$ such that $\operatorname {tr}(AB) = \operatorname {tr}(A)\operatorname {tr}(B)$?

Given a $2 \times 2$ matrix $M$ over $\mathbb {C}$. What can be said about the determinant of $M \otimes I_{2} - I_{2} \otimes M$?

Let $A$, $B$ be $n \times n$ matrices over $\mathbb {C}$. Then $\operatorname {tr}([A,B]) = 0$. Consider $n=3$ and

with $\operatorname {tr}(C) = 0$. Construct $3 \times 3$ matrices $A$ and $B$ such that $[A,B] = C$.

Consider the $k$ vectors

in the Euclidean space $\mathbb {R}^{n}$. The parallelepiped determined by these vectors has the $k$-dimensional area

where $V$ is the $n \times k$ matrix with $\overrightarrow {v}_{1}, \overrightarrow {v}_{2}, \ldots , \overrightarrow {v}_{k}$ as its columns. Apply it to the two vectors in $\mathbb {R}^{3}$

The equation of a hyperplane passing through the $n$ points $\overrightarrow {x}_{1}, \overrightarrow {x}_{2}, \ldots , \overrightarrow {x}_{n}$ in the Euclidean space $\mathbb {R}^{n}$ is given by

Let $n=3$. Apply it to the three vectors in $\mathbb {R}^{3}$

Let $A$, $B$ be $2 \times 2$ matrices. Show that

Can this identity be extended to $3 \times 3$ matrices?

Let $A$, $B$ be $n \times n$ matrices. Let $U$, $V$ be unitary $n \times n$ matrices. Show that

applying cyclic invariance and $UU^{*} = I_{n}$, $VV^{*} = I_{n}$.

Let $A$, $B$ be $2 \times 2$ matrices over $\mathbb {C}$ with $\operatorname {tr}(A) = 0$ and $\operatorname {tr}(B) = 0$. Show that the condition on $A$ and $B$ such that $\operatorname {tr}(AB) = 0$ is given by

Let $A$, $B$ be nonzero $2 \times 2$ matrices over $\mathbb {R}$ with $\operatorname {det}\left(A \otimes B\right) = 0$. Can we conclude that both $A$ and $B$ are noninvertible?

Consider the symmetric $n \times n$ band matrix ($n \geq 3$)

with the elements in the field $\mathbb {F}_{2}$. Show that

with the initial conditions $\operatorname {det}\left(M_{3}\right) = \operatorname {det}\left(M_{4}\right) = 1$. Show that the solution is

Consider the nonnormal matrices

Find $(\operatorname {det}\left(A_{2}^{*}A_{2}\right))^{1/2}$ and $(\operatorname {det}\left(A_{3}^{*}A_{3}\right))^{1/2}$. Extend to $n$ dimensions.

Let $A$ be a $n \times n$ matrix with $\operatorname {det}\left(A\right) = -1$. Show that $\operatorname {det}\left(A^{-1}\right) = -1$.

Let $n$ be even. Consider a skew-symmetric $n \times n$ matrix $A$ over $\mathbb {R}$. Let $B$ be a symmetric $n \times n$ matrix over $\mathbb {R}$ with the entries of $B = (b_{jk})$ given by $b_{jk} = b_{j}b_{k}$ ($b_{j}, b_{k} \in \mathbb {R}$). Show that

Notice that $\operatorname {det}\left(B\right) = 0$. Let $S_{n}$ be the symmetric group. Then we have

Let $h>0$ and $b>a$. Consider the matrix

Show that $\operatorname {det}\left(M\right) = h(b-a) > 0$ and the inverse of $M$ is given by

The $n \times n$ permutation matrices form a group under matrix multiplications. Show that $\operatorname {det}\left(I_{n}-P\right) = 0$ for any $n \times n$ permutation matrix.

Let $\overrightarrow {u}, \overrightarrow {v}, \overrightarrow {w}$ be column vectors in $\mathbb {R}^{3}$. We form the $3 \times 3$ matrix

Show that $\operatorname {det}\left(M\right) = (\overrightarrow {u} \times \overrightarrow {v}) \cdot \overrightarrow {w}$.

Consider the $2 \times 2$ matrix

Show that $\operatorname {tr}(C^{k}) = \operatorname {tr}(C^{k-1}) + \operatorname {tr}(C^{k-2})$ with $k=3,4,\ldots$

Let $A$ be an $n \times n$ matrix. Assume that

Can we conclude that $\operatorname {det}\left(A\right) = 0$?

Let $A_{1}$, $A_{2}$, $A_{3}$ be $2 \times 2$ matrices over $\mathbb {R}$. We define

Let $A$, $B$ be $n \times n$ matrices over $\mathbb {C}$. Assume that $B$ is invertible. Show that $\operatorname {det}\left(I_{n}+BAB^{-1}\right) = \operatorname {det}\left(I_{n}+A\right)$.

An $n \times n$ matrix $A$ is called idempotent if $A^{2} = A$. Show that $\operatorname {rk}\left(A\right) = \operatorname {tr}(A)$.

Let $A$ be a real symmetric and positive definite $n \times n$ matrix. Then

For the identity matrix $I_{n}$ we obtain equality. Show that for the matrix

we have $n=2$, $\operatorname {det}\left(A\right) = 2$, $\operatorname {tr}(A) = 3$ and $\ln (2) < 1$.

Let $A$, $B$, $C$ be $n \times n$ matrices over $\mathbb {C}$. Let $\overrightarrow {a}_{j}$, $\overrightarrow {b}_{j}$, $\overrightarrow {c}_{j}$ ($j=1,\ldots ,n$) be the $j$-th column of $A$, $B$, $C$, respectively. Show that if for some $k \in \left\{ 1,\ldots ,n\right\}$

and

then $\operatorname {det}\left(C\right) = \operatorname {det}\left(A\right)+\operatorname {det}\left(B\right)$.

Consider the Legendre polynomials $p_{j}$ ($j=0,1,\ldots$) with

Show that

Note that ($x$ omitted)

and $p_{0}(0) = 1$, $p_{1}(0) = 0$, $p_{2}(0) = -1/2$, $p_{3}(0) = 0$, $p_{4}(0) = 3/8$.

Let $A$ be an $n \times n$ diagonal matrix over $\mathbb {C}$. Let $B$ be an $n \times n$ matrix over $\mathbb {C}$ with $b_{jj} = 0$ for all $j=1,\ldots ,n$. Can we conclude that all diagonal elements of the commutator $[A,B]$ are $0$?

Let $A$, $B$ be $n \times n$ matrices over $\mathbb {R}$ with $\operatorname {det}\left(A\right) = 1$ and $\operatorname {det}\left(B\right) = 1$. This means $A$, $B$ are elements of the Lie group $SL(n,\mathbb {R})$. Can we conclude that

Let $H$ be a hermitian $n \times n$ matrix. Show that $\operatorname {det}\left(H+iI_{n}\right) \neq 0$.

Let $A$ be an $2 \times 2$ matrix over $\mathbb {R}$. Calculate $r = \operatorname {tr}(A^{2}) - (\operatorname {tr}(A))^{2}$. Show that the conditions on $a_{jk}$ such that $r=0$ are given by $a_{12}a_{21} = a_{11}a_{22}$, i.e. $\operatorname {det}\left(A\right) = 0$.

Consider a triangle embedded in $\mathbb {R}^{3}$. Let $\overrightarrow {v}_{j} = (x_{j}, y_{j}, z_{j})$ ($j=1,2,3$) be the coordinates of the vertices. Then the area $A$ of the triangle is given by

where $\times$ denotes the vector product and $\left\| \cdot \right\|$ denotes the Euclidean norm. The area $A$ of the triangle can also be found via

where $D$ denotes the determinant. Consider

Find the area of the triangle using both expressions. Discuss. The triangle could be one of the faces of a tetrahedron.

Consider the $3 \times 3$ matrix $M$ with entries

Find the determinant of this matrix.

The permanent and the determinant of an $n \times n$ matrix $M$ over $\mathbb {C}$ are respectively defined as

where $S_{n}$ denotes the symmetric group on a set of $n$ symbols. For an $n \times n$ matrix $A$ and an $m \times m$ matrix $B$ we know that

Study $\operatorname {perm}(A \otimes B)$.

Let $A$, $B$, $C$, $D$ be $n \times n$ matrices. Assume that $D$ is invertible. Consider the $(2n) \times (2n)$ matrix

Show that $\operatorname {det}\left(M\right) = \operatorname {det}\left(AD-BD^{-1}CD\right)$ using the identity

Let $\mu \in \mathbb {R}$ and $A$ be an $2 \times 2$ matrix over $\mathbb {R}$. Show that the determinant of the $4 \times 4$ matrix

only contains even powers of $\mu$.

Consider the invertible $3 \times 3$ matrix $A$

Find a permutation matrix $P$ such that $PA^{-1}P^{-1} = A$.

Let $x, \epsilon \in \mathbb {R}$. Show that the determinant of the symmetric $n \times n$ matrix

is given by $\operatorname {det}\left(A(x,\epsilon )\right) = \epsilon^{n-1}(nx+\epsilon )$.

Show that the determinant of the $2n \times 2n$ matrix

is given by $\operatorname {det}\left(\Omega \right) = \left(\operatorname {det}\left(\begin{array}{ll} 0 & 1 \\ -1 & 0 \end{array}\right)\right)^{n} (\operatorname {det}\left(I_{n}\right))^{2} = 1$.

Given a $2 \times 2 \times 2$ hypermatrix $A = (a_{jk\ell })$, $j,k,\ell = 0,1$ and the $2 \times 2$ matrix

The multiplication $AS$ which is again a $2 \times 2$ hypermatrix is defined by

Assume that $\operatorname {det}\left(S\right) = 1$, i.e. $S \in SL(2,\mathbb {C})$. Show that $\operatorname {Det}(AS) = \operatorname {Det}(A)$. This is a typical problem to apply computer algebra.

The Levi-Civita symbol (also called completely antisymmetric constant tensor) is defined by

Let $\delta_{jk}$ be the Kronecker delta. Show that