2

What would the inverse of the following functions be?

\begin{align*} f(x,y,z)&:= (x-3y + 5z -14, x-4y+5z-3, 2x-3y+4z+1) \\ g(x,y)&:= (5x-3y-22, 2x - 2y -9) \end{align*}

I would normally do these by using the matrix system, and by applying the Gauss-Jordan method I'd get the identity matrix and the inverse. But since I've got loose numbers with no variables, I don't know how to do this.

And how would i be able to check if the answer is right once i got it?

Thanks

icurays1
  • 17,161
user143899
  • 189

2 Answers2

2

The way to deal with the loose numbers is to just bring them out as a constant. Let me explain what I mean. Notice that $f$ satisfies: $$ f(x, y, z) = \begin{bmatrix} 1 & -3 & 5 \\ 1 & -4 & 5 \\ 2 & -3 & 4 \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ \end{bmatrix} + \begin{bmatrix} -14 \\ -3 \\ 1 \\ \end{bmatrix} $$

In other words, if $u = (x,y,z)$, we can write $f(u) = Au + u_0$, where $A$ is a matrix and $u_0$ is a constant vector. Then the inverse of $f$ will just be $$ f^{-1}(v) = A^{-1}(v - u_0) $$ You can check that this is the inverse by plugging in $f(u)$ for $v$. So you can find the inverse function if you know the inverse of the matrix, as usual.

Caleb Stanford
  • 45,895
1

Hint: what is the inverse of $y=mx+b$? Well, we would have $y-b=mx$, and so $x=$...

Your situation is just like this, but you have $y=Ax+b$.

icurays1
  • 17,161