Part 1

Compute the following matrix products
  1. Product: $$ \begin{bmatrix} -2 & 1 & 3\\ 5 & 0 & 2 \end{bmatrix} \cdot \begin{bmatrix} 4 & 2 & 5\\ -3 & -1 & 1\\ 0 & 2 & 3 \end{bmatrix} = $$
  2. Product: $$ \begin{bmatrix} 3 & -1 \\ 5 & 2 \\ -1 & 1 \end{bmatrix} \cdot \begin{bmatrix} 4 & 2 \\ -3 & 1 \end{bmatrix} = $$
  3. Product: $$ \begin{bmatrix} 3 & -1 \\ 5 & 2 \\ -1 & 1 \end{bmatrix} \cdot \begin{bmatrix} 4 & 2 \\ -3 & 1 \\ 2 & 3 \end{bmatrix} = $$
  4. Product: $$ \begin{bmatrix} -2 & -1 \\ 3 & 2 \end{bmatrix} \cdot \begin{bmatrix} -3 & 1 \\ 2 & 5 \end{bmatrix} = $$

Part 2

  1. Make up two 2x2 matrices, $A$ and $B$, with random (small) values. Calculate $A\cdot B$ and $B\cdot A$, and compare the results. What did you just prove?
  2. For square matrices, i.e. matrices of dimension $n\times n$, is there a matrix that serves as the additive identity? What is it for 3x3 matrices?
  3. Recall our definition of the unit vectors:
    The $i$th unit vector in vector space $R^n$, denoted $\boldsymbol{e_i}$, is the vector that is $1$ in its $i$th component and zero everywhere else. If it is not clear by context, we will specify whether it is a row or column vector.
    and the identity matrix:
    The $n\times n$ identity matrix, denoted by $I_n$, is the matrix whose $i$th row is $\boldsymbol{e_i}$, the $i$th unit (row) vector. Note that this also means that the $i$th column is the $i$th unit (column) vector.
    Compute $$ I_3 \cdot \left[\begin{array}{ccc} a&b&c\\ d&e&f\\ g&h&i \end{array}\right] $$ and verify that $I_3$ really is the multiplicative identity for 3x3 matrices. Note: be sure to write out the matrix $I_3$.
    In fact, we have the following theorem:

    If $A$ is an $n\times n$ matrix, then $I_n\cdot A = A$.