# Question #4ca6c

Oct 24, 2016

$x = - \frac{1}{2}$
$y = 33$

#### Explanation:

Let $A = \left[\begin{matrix}4 & 1 & 3 \\ 5 & 2 x & - 3\end{matrix}\right]$, $B = \left[\begin{matrix}9 & 3 \\ - 6 & 2 \\ 1 & 4\end{matrix}\right]$, $C = \left[\begin{matrix}y & - 3 \\ 45 & 2\end{matrix}\right]$

Using the properties of matrix multiplication, we know that

$y = {c}_{11}$

$= {a}_{11} {b}_{11} + {a}_{12} {b}_{21} + {a}_{13} {b}_{31}$

$= 4 \left(9\right) + 1 \left(- 6\right) + 3 \left(1\right)$

$= 36 - 6 + 3$

$= 33$

Similarly, we know that

$2 = {c}_{22}$

$= {a}_{21} {b}_{12} + {a}_{22} {b}_{22} + {a}_{23} {b}_{32}$

$= 5 \left(3\right) + 2 x \left(2\right) + \left(- 3\right) \left(4\right)$

$= 15 + 4 x - 12$

$= 4 x + 3$

$\implies 4 x = 2 - 3 = - 1$

$\implies x = - \frac{1}{2}$

Notice that we didn't need to do all the multiplication. We only needed to multiply rows and columns to get equations containing our variables.

(Also, as a side note, if we did do the full multiplication, we'd find that the book has a mistake, in that ${c}_{12}$ should be $26$, rather than $- 3$)