Identity matrix

Let $A \in \mathcal{M}_3 \left( \mathbb{R} \right)$ such that $\det A =1$ and ${\rm tr} (A)= {\rm tr} (A^{-1})=0$. Prove that $A^3=\mathbb{I}_{3 \times 3}$.

Solution

Linear map and trace

Let $f:\mathbb{F}^{n \times n} \rightarrow \mathbb{F}$ be a linear map such that $f\left ( AB \right ) = f \left ( BA \right )$  forall $A, B \in \mathbb{F}^{n \times n}$. Prove that there exists a $\kappa \in \mathbb{F}$ such that $f\left ( A \right ) = \kappa \;{\rm tr} \left ( A \right )$ forall $A \in \mathbb{F}^{n \times n}$.

Solution

A zero determinant

Let $A \in \mathcal{M}_n \left( \mathbb{C} \right)$ with $n \geq 2$ such that

$$\det \left ( A+X \right )=\det A + \det X$$

for every matrix $X \in \mathcal{M}_n \left( \mathbb{C} \right)$. Prove that $A=\mathbb{O}$.

Solution

On a determinant

Let $p$ be a prime number and let $\omega$ be a primitive $p$-th root of unity. Define:

$$\mathcal{V} = \det \begin{pmatrix}
1 &1  &1  &\cdots  &1 \\
 1& \omega &\omega^2  &\cdots  &\omega^{p-1} \\
1 &\omega^2  &\left ( \omega^2 \right )^2  &\cdots  &\left ( \omega^2 \right )^{p-1} \\
 \vdots&\vdots  &\vdots  &\ddots  & \vdots\\
 1& \omega^{p-1} &\left ( \omega^{p-1} \right )^2  &\cdots  & \left ( \omega^{p-1} \right )^{p-1}
\end{pmatrix}$$

Evaluate the rational number $\mathcal{V}^2$.

Solution

A non diagonizable matrix

Let $A \in \mathcal{M}_{2 \times 2} (\mathbb{R})$ such that $A^2+\mathbb{I}_{2 \times 2}=0$. Evaluate the minimal polynomial of $A$ and prove that $A$ is not diagonizable.

Solution

Invertible matrices

Consider the matrices $A \in \mathcal{M}_{m \times n}$ and $B \in \mathcal{M}_{n \times m}$. If $AB +\mathbb{I}_m$ is invertible prove that $BA+\mathbb{I}_n$ is also invertible.

Solution

A determinant

Let $\gcd(i, j)$ denote the greatest common divisor of $i, j$ and let $\varphi$ denote Euler's totient function. Prove that:

$$\begin{vmatrix}
\gcd(1,1) &\gcd(1, 2)  &\cdots  & \gcd(1,n)\\
 \gcd(2,1)&\gcd(2,2)  &\cdots  & \gcd(2,n)\\
 \vdots&  \vdots  & \ddots &\vdots \\
 \gcd(n,1)&\gcd(n,2)  &\cdots  &\gcd(n,n)
\end{vmatrix}= \prod_{j=1}^{n}\varphi(j)$$

Solution

A value of a determinant

Let $A \in \mathcal{M}_n(\mathbb{R})$ such as $A^3=4I_n-3A$.Prove that $$\det(A+I_n)=2^n$$

Solution

A determinant

Let $a_i>0$. Evaluate the determinant:

$$\left[\begin{matrix}a_1&a_2&a_3&\ldots&\ldots&a_{n-1}&a_n\\-x&x&0&\ldots&\ldots&0&0\\0&-x&x&\ldots&\ldots&0&0\\\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\0&0&0&\ldots&\ldots&x&0\\0&0&0&\ldots&\ldots&-x&x\end{matrix}\right]$$

Solution

There do not exist points

Prove that there do not exist four points in $\mathbb{R}^2$ whose pairwise distances are all odd integers.

Solution

Positive determinant of a matrix

Let $A \in \mathbb{R}$ be a matrix  such that $A^3=A +\mathbb{I}$. Prove that $\det A >0$.

Solution

Equal determinants

Let $A, B \in \mathbb{R}^{n \times n}$ that are diagonizable in $\mathbb{R}$. If $\det (A^2+B^2)=0$ and $AB=BA$ , then prove that $\det A = \det B =0$.

Solution

Direct sum

We know that every function can be written as the sum of an odd and even function. That is:

$$f(x)={\rm E}+ {\rm O}$$

where ${\rm E}$ is the even function which is no other than  ${\rm E}=\frac{f(x)+f(-x)}{2}$ and the odd function is no other than ${\rm O} = \frac{f(x)-f(-x)}{2}$. Also this decomposition is unique (left as an exercise to the reader). Prove that the sum of the odd and the even function is a direct sum.

Solution

Determinant and eigenvalues of orthogonal matrix

Let $A$ be an orthogonal matrix . Prove that:

a) The determinant of the matrix $A$ is $\det A = \pm 1$.
b) The absolute value of the eigenvalues (if there exist) of the matrix $A$ is $1$.

Solution

Rotation matrix

Give the matrix

$$A= \begin{pmatrix}
\sqrt{3}/2 &0  &1/2 \\
 0& 1 &0 \\
 -1/2& 0 & \sqrt{3}/2
\end{pmatrix}$$

examine if it is a rotation of a plane around an axis that is perpendicular to it. If so, determine the angle of rotation and the axis.

Solution

Perpendicular eigenvectors

Let $A \in \mathbb{R}^{n \times n}$ be a matrix such that $A=A^t$. Prove that, if $u$ and $v$ are eigenvectors of $A$ corresponding to different eigenvalues , then $\langle u, v \rangle =0$.

Solution

Minimal polynomial and zero matrix

Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix. Suppose that there exists an integer $m>0$ such that $A^m = \mathbb{O}_{n \times n}$. Prove that $A$ is the zero matrix.

Solution

Symmetric Matrix

Let $A\in \mathbb{R}^{n \times n}$ be a symmetric matrix such that $A^3=A^2$. Prove that $A^2=A$.

Solution

Non invertible matrix

Prove that for all matrices $A \in \mathbb{R}^{2\times 2}$ there are at most two values of $\ell \in \mathbb{R}$ such that the matrix $A+\ell \mathbb{I}_2$ is not invertible.

Solution

Trace of matrix and Cayley Hamilton

Let $A \in \mathbb{R}^{2 \times 2}$. Prove that:
$$A^2-\text{tr}\left(A\right)A+\det\left(A\right) \mathbb{I}=0$$

Solution