Worksheet 1

1. What is the coefficient matrix and the augmented matrix for the following system of linear equations: \(x_1 + x_2 + x_3 = 1 \\ 2x_1 + 3x_2 + x_3 = 2 \\ x_1 - x_2 + 3x_3 = 2.\)

2. Solve the system given in Problem 1 using the algorithm described in the notes

3. For the matrix \(\begin{pmatrix} 1 & -2 & 0 & 1 \\ 3 & 0 & 4 & -1 \\ 5 & 5 & 5 & 5 \end{pmatrix}\) write out the set of column vectors and row vectors.

4. Compute the transpose of the matrix in Problem 3.

5. For the matrices $ A = \begin{pmatrix} 1 & 0 & 1 & 0 \\ 1 & 1 & -1 & -1 \\ 0 & 0 & 1 & 1 \end{pmatrix} \ , \ B = \begin{pmatrix} -3 & -1 & 0\\ 2 & 0 & 0 \\ 1 & 1 & 0 \end{pmatrix} \ , \ \mathbf{v} = \begin{pmatrix} 1 & 1 & 1 \\ \end{pmatrix} \ , \ \mathbf{w} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} $

   Compute all the products (as matrices) that are well-defined. Which ones are not well-defined?

6. Which of the following statements are true?
   • For any two matrices $A$ and $B$ of the same size, we always have $A + B = B + A$
   • For any two matrices $A$ and $B$, we always have $AB = BA$.
   • For any two square matrices $A$ and $B$ of the same size, we always have $AB = BA$.
7. For any three matrices $A$, $B$, and $C$, either
   • both $(AB)C$ and $A(BC)$ are not well-defined or
   • $(AB)C = A(BC)$.

   Can you convincingly explain why? A professional mathematician would say matrix multiplication is associative.

8. We have the following problem: we want to find a cubic polynomial \(p(x) = a + bx + cx^2 + dx^3\) such that $p(1) = 5$, $p^\prime(1) = 5$, $p(2) = 17$, and $p^\prime(2) = 21$. What does this have to do with linear algebra?