# How do you find missing values in a matrix?

## The matrix I have is given by $M = \left(\begin{matrix}1 & a \\ 0 & 2\end{matrix}\right)$, and the image of the point $\left(b , 5\right)$ under $M$ is $\left(5 , b\right)$. I need to find the values of $a$ and $b$.

Apr 5, 2018

$\textcolor{b l u e}{a = - 1}$

$\textcolor{b l u e}{b = 10}$

#### Explanation:

We have:

$\boldsymbol{M} = \left[\begin{matrix}1 & a \\ 0 & 2\end{matrix}\right]$

$\boldsymbol{A} = \left[\begin{matrix}b \\ 5\end{matrix}\right]$

$\boldsymbol{B} = \left[\begin{matrix}5 \\ b\end{matrix}\right]$

We are told:

$\boldsymbol{M A} = \boldsymbol{B}$

$\boldsymbol{M A} = \left[\begin{matrix}1 & a \\ 0 & 2\end{matrix}\right] \left[\begin{matrix}b \\ 5\end{matrix}\right] = \left[\begin{matrix}b + 5 a \\ 10\end{matrix}\right]$

$\therefore$

$\left[\begin{matrix}b + 5 a \\ 10\end{matrix}\right] = \boldsymbol{B} = \left[\begin{matrix}5 \\ b\end{matrix}\right]$

$\left[\begin{matrix}b + 5 a \\ 10\end{matrix}\right] = \left[\begin{matrix}5 \\ b\end{matrix}\right]$

$\therefore$

$b + 5 a = 5 \textcolor{w h i t e}{88} \left[1\right]$

$10 = b \textcolor{w h i t e}{88} \left[2\right]$

So $b = 10$

Substituting $b = 10$ in $\left[1\right]$

$10 + 5 a = 5 \implies a = - 1$

$\textcolor{b l u e}{a = - 1}$

$\textcolor{b l u e}{b = 10}$