I am about to start a PhD in zoology. At the moment, I am looking at matrices, as I have never looked at them before, and will need some further maths during my PhD.
I have found a question in the textbook where I need to solve simultaneous equations using matrix algebra.
$$ \begin{aligned} 2x + 3y &= 3 \\ 6x + 5y &= 0 \end{aligned} $$
Usually, I am quite good at figuring these out, for example, in an integral or differential situation. However, I don't even know where to get started when "using matrix algebra" is concerned. If anybody could offer me the tiniest piece of guidance I'd be grateful.
The question may be formulated as $$ \begin{bmatrix}2&3\\6&5\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} =\begin{bmatrix}3\\0\end{bmatrix} $$ You solve it by finding the inverse of the leftmost matrix and multiplying by that inverse from the left to obtain $$ \begin{bmatrix}x\\y\end{bmatrix} =\begin{bmatrix}2&3\\6&5\end{bmatrix}^{-1}\begin{bmatrix}3\\0\end{bmatrix} $$ which means that $\left[\begin{smallmatrix}x\\y\end{smallmatrix}\right]$ may be found by calculating the product on the right-hand side.