I'm really stuck on these linear transformations, so I have $T(x_1,x_2)=(-5x_1+9x_2,4x_1-7x_2)$, and I need to show that $T$ is invertible. So would I pretty much just say that this is the matrix: $$\left[\begin{matrix}-5&9\\4&-7\end{matrix}\right]$$ Then it's inverse must be $\frac{1}{(-5)(-7)-(9)(4)}\left[\begin{matrix}-7&-9\\-4&-5\end{matrix}\right]=\left[\begin{matrix}7&9\\4&5\end{matrix}\right]$. But is that "showing" that $T$ is invertible? I'm also supposed to find a formula for $T^{-1}$. But that's the matrix I just found right?
[Math] How to show that $T$ is invertible
linear algebramatrices
Best Answer
A fast way to check if a matrix is invertible, is to calculate $\det(T)$. If it's equal to $0$, you can't invert $T$, otherwise you can.
To find the general formula to invert a $2 \times 2$ matrix, try inverting one with $a,b,c,d$ as elements.
Spoiler alert : $$ \begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1}= \frac{1}{\det(T)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} $$