Let’s analyze each question in the image and outline how to approach them.

Question Breakdown and Solutions:

1. Compute the matrix products $AB$ and $BA$:

   Given: $A = \begin{bmatrix} a_1 & a_2 & \dots & a_n \end{bmatrix}$ and $B = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix}.$

   • For $AB$: This is the product of a $1 \times n$ row vector $A$ and an $n \times 1$ column vector $B$, resulting in a $1 \times 1$ scalar. The result will be $a_1b_1 + a_2b_2 + \dots + a_nb_n$.

   • For $BA$: This product is the multiplication of an $n \times 1$ column vector $B$ with a $1 \times n$ row vector $A$, resulting in an $n \times n$ matrix. The $i,j$-th element will be $b_i a_j$.

2. Consider the matrix $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$. Compute $A^2$, $A^3$, and $A^4$, and conjecture a formula for $A^n$:

   • To compute $A^2$: Multiply $A$ by itself.
   • Similarly, for $A^3$ and $A^4$, use the results from previous powers.
   • Conjecture a formula: Once the patterns are observed, you can propose a general form for $A^n$, which might involve a multiple of a matrix with all entries equal. Then, prove the formula using mathematical induction.

3. Let $A$ be a nilpotent matrix. Show that there exists a matrix $B$ such that $B(I + A) = (I + A)B = I$:

   • Since $A$ is nilpotent, there exists a power $k$ such that $A^k = 0$.
   • Construct $B$ as a finite series that cancels out the effect of $A$, likely as $B = I - A + A^2 - \dots$ up to the term that nullifies $A$.
   • Verify that $B(I + A) = I$.

4. Solve the system of linear equations: $x + 2y - z = 10$ $-x + y + 2z = 2$ $2x + y - 3z = 2$

   • Use methods such as Gaussian elimination or matrix methods to find values of $x$, $y$, and $z$.
   • Check if the system has a unique solution, infinite solutions, or no solutions.

5. Let $A$ and $B$ be $m \times n$ matrices and $x$ be an $n \times 1$ column matrix. Prove that if $Ax = 0$ for all $n \times 1$ column matrices $x$, then $A$ is a zero matrix. Prove that if $Ax = Bx$ for all $x$, then $A = B$:

   • For the first part, show that $Ax = 0$ for all $x$ implies that all entries in $A$ must be zero, using properties of matrix-vector multiplication.
   • For the second part, if $Ax = Bx$ for all $x$, conclude that $A = B$ by considering the matrix elements directly and showing that they must be identical.

Would you like a detailed solution for each part, or do you have specific questions on any of these?

Here are 5 further questions to expand on these topics:

1. How would you generalize the product $AB$ for matrices of arbitrary sizes?
2. Can you find an explicit formula for $A^n$ in question 2 without using induction?
3. Why is it significant that $A$ is nilpotent in question 3? How does that influence the solution?
4. Could you solve the linear system in question 4 using matrix inverses, and under what conditions is that possible?
5. What properties of matrix multiplication ensure that $Ax = Bx$ for all $x$ implies $A = B$?

Tip: When solving a system of equations, always check for consistency before attempting to find a unique solution—this can save time if the system turns out to be inconsistent.