Exercises

1. Consider a system of linear questions

   $$\begin{equation*} (a_1 + b)x_1 + a_2 x_2 + a_3 x_3 + \cdots + a_n x_n = 0 \end{equation*}$$ $$\begin{equation*} a_1 x_1 + (a_2 + b)x_2 + a_3 x_3 + \cdots + a_n x_n = 0 \end{equation*}$$ $$\begin{equation*} a_1 x_1 + a_2 x_2 + (a_3 + b)x_3 + \cdots + a_n x_n = 0 \end{equation*}$$ $$\begin{equation*} \vdots \end{equation*}$$ $$\begin{equation*} a_1 x_1 + a_2 x_2 + a_3 x_3 + \cdots + (a_n + b)x_n = 0 \end{equation*}$$

   where $\sum_{i=1}^n a_i \ne 0$. Determine the relationship between $a_1, a_2, \cdots, a_n$ and $b$ such that

   a. This linear system has only a zero solution.

   b. This system has nonzero solutions. In this case, determine a solution set.

2. Prove

   $$\begin{equation*} \operatorname{rank}(L_A) + \operatorname{rank}(L_B) - n \;\le\; \operatorname{rank}(L_{AB}) \;\le\; \min\{\operatorname{rank}(L_A), \operatorname{rank}(L_B)\}, \end{equation*}$$

   where $A$ and $B$ are $n \times n$ matrices.

3. Prove

   $$\begin{equation*} \operatorname{rank}(L_{A+B}) \;\le\; \operatorname{rank}(L_A) + \operatorname{rank}(L_B) \end{equation*}$$

   where $A$ and $B$ are $n \times m$ matrices.

4. Let $A, B \in M_{n \times n}(\mathbb{R})$. Suppose $AB = BA = 0$ and $\operatorname{rank}(A^2) = \operatorname{rank}(A)$. Prove that

   $$\begin{equation*} \operatorname{rank}(A + B) = \operatorname{rank}(A) + \operatorname{rank}(B) \end{equation*}$$

5. Let $A, B, C, D$ be $n \times n$ matrices such that they commute pairwise under matrix multiplication, and $AC + BD = I_n$. Prove that:

   $$\begin{equation*} \operatorname{rank}(AB) = \operatorname{rank}(A) + \operatorname{rank}(B) - n. \end{equation*}$$

6. $A$ is an $n \times n$ real matrix, $AA^t = k^2 I_n$, $k \in \mathbb{R}$. Prove:

   $$\begin{equation*} \operatorname{rank}(k I_n - A) = \operatorname{rank}\big((k I_n - A)^2\big). \end{equation*}$$

7. Suppose $A$ is an $n \times n$ matrix with rank $r$ ($1 \le r \le \frac{n}{2}$), and $A^2 = 0$. Prove that there exists an invertible matrix $P$ such that

   $$\begin{equation*} A = P \begin{pmatrix} 0 & I_r & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} P^{-1} \end{equation*}$$

   where $I_r$ is the $r \times r$ identity matrix, and $0$ are zero matrices.

8. If $A^2 = I_n$, Prove $A$ is diagonalizable (here $A$ is a real square matrix).

9. Let $V$ be a finite-dimensional real inner product space, and let $U$ and $T$ be self-adjoint linear operators on $V$ such that $UT = TU$.

   Prove that there exists an orthonormal basis of $V$ consisting of vectors that are eigenvectors of both $U$ and $T$.

10. Let $A$ and $B$ be $n \times n$ complex matrices satisfying the condition $AB = A + B.$

    Prove that $A$ and $B$ are simultaneously triangularizable (there exists a basis under which both $A$ and $B$ are upper triangular matrices).

11. Let $A$ be an $n \times n$ matrix satisfying the polynomial equation $\sum_{i=0}^{m} b_i A^i = 0$, where the coefficients satisfy the condition $|b_m| > \sum_{i=0}^{m-1} |b_i|.$

    Prove that the equation $2X - XA^2 = AX$ has only the trivial solution $X = 0$.