Perform each of the following matrix operations without using a calculator. If the operation is impossible, explain why.

1. $\left[ \begin{array} { l l l } { 4 } & { 9 } & { 2 } \\ { 6 } & { 0 } & { 5 } \end{array} \right] \left[ \begin{array} { l } { a } \\ { b } \\ { c } \end{array} \right]$ 

   Hint (a):

   Multiplying a $2 \times 3$ matrix by a $3 \times 1$ matrix results in a $2 \times 1$ matrix.

2. $2 \left[ \begin{array} { r r r } { 4 } & { 9 } & { 2 } \\ { 6 } & { 0 } & { - 5 } \end{array} \right] + \left[ \begin{array} { r r r } { 1 } & { 0 } & { - 3 } \\ { 0 } & { 4 } & { 1 } \end{array} \right]$ 

   Step (b):

   $\begin{bmatrix}8&18&4\\12&0&-10\end{bmatrix}+\begin{bmatrix}1&0&-3\\0&4&1\end{bmatrix}$

3. $\left[ \begin{array} { l l } { a } & { b } \\ { c } & { d } \end{array} \right] \left[ \begin{array} { l l } { e } & { f } \end{array} \right]$ 

   Hint (c):

   Is it possible to multiply a $2 \times 2$ matrix by a $1 \times 2$ matrix?

4. $\left[ \begin{array} { l l } { a } & { b } \\ { c } & { d } \end{array} \right] + \left[ \begin{array} { l l } { e } & { f } \end{array} \right]$ 

   Hint (d):

   Is it possible to add two matrices with different dimensions?

5.  $\begin{vmatrix} 10 & 1 \\ -9 & 2 \\ \notag \end{vmatrix}$ 

   Hint (e):

    $\begin{vmatrix} a & b \\ c & d \\ \notag \end{vmatrix} = ad-cb$

6.  $\begin{vmatrix} 4 & 2 & 5 \\ 10 & 3 & -6 \\ \notag \end{vmatrix}$ 

   Hint (f):

   See the hint in part (e). Is this a square matrix?