How can I solve $Ax=y$ for nonsquare matrix $A$.

Question

Say I have some nonsquare matrix $A$ and a vector $y$, how can I solve $Ax=y$? If $A$ was square, it would be simply $A^{-1}y=x$, but you can't invert a nonsquare matrix.

Answer 1

In general, you can't invert (square) matrices. Look up "Properties of Invertible Matrices" to see what conditions must be satisfied for a matrix to be invertible.

A matrix has to be square to b e invertible, but this does not necessarily hold the other way around.

Gaussian elimination is, I would say, the most used process to determine solutions to the equation

$$A \mathbf{x} = \mathbf{b}$$

where:

$A$ is an $m \times n$ matrix, also written as $A_{m \times n}$ or $A \in \mathbb{M}^{m \times n}$

$\mathbf x$ is an $n \times 1$ vector, also written as $\mathbf{x} \in \mathbb{R}^n$

$\mathbf{b}$ is a $m \times 1$ vector, also written as $\mathbf{b} \in \mathbb{R}^m$