How do you multiply ((4, 0), (-1, 3), (2, -5)) with ((1),( -3))?

Dec 14, 2016

$\left[\begin{matrix}4 \\ - 10 \\ 17\end{matrix}\right]$

Explanation:

$\textcolor{w h i t e}{a a a} \left[A\right] \textcolor{w h i t e}{a a a a a a a} \left[B\right] \textcolor{w h i t e}{a a a a a} \left[C\right]$

$\left[\begin{matrix}4 & 0 \\ - 1 & 3 \\ 2 & - 5\end{matrix}\right] \cdot \left[\begin{matrix}1 \\ - 3\end{matrix}\right] = \left[\begin{matrix}a \\ b \\ c\end{matrix}\right]$

$\textcolor{w h i t e}{a a} \textcolor{red}{3} \text{ x "color(blue)2color(white)(aaaa)color(blue)2" x "color(red)1color(white)(aa)color(red)3" x } \textcolor{red}{1}$

The size of the first matrix $A$ is $\text{2 x 2}$ because it has 2 rows and 2 columns. The size of the second matrix $B$ is $\text{2 x 1}$ because it has 2 rows and 1 column.

When the sizes are written next to each other, the "inner" numbers must match. The "outer" numbers determine the size of the resultant matrix.

The resultant matrix $C$ has been labeled $\left[\begin{matrix}a \\ b \\ c\end{matrix}\right]$ and has size $3 \text{ x } 1$.

Position $a$ is in the first row and first column. It's value is obtained by multplying the first row of matrix $A$ by the first column of matrix $B$.

$a = 4 \cdot 1 + 0 \cdot 3 = 4$

Position $b$ is in the second row and first column. It is the result of multiplying the second row of $A$ by the first column of $B$.

$b = - 1 \cdot 1 + 3 \cdot - 3 = - 10$

Position $c$ is obtained by multiplying the third row of $A$ by the first column of $B$.

$c = 2 \cdot 1 + - 5 \cdot - 3 = 17$

The answer is then $\left[\begin{matrix}4 \\ - 10 \\ 17\end{matrix}\right]$

Dec 14, 2016

Shell wrote a good answer while I was writing mine.