How to test whether a matrix transformation is one-to-one, onto, or invertible
A matrix transformation can fail to have an inverse in two ways. Different inputs can land on the same output, or some outputs can be missed. Row reduction tells us which of those can happen by showing where the pivots are.
Use a pivot in every column for one-to-one
A map is one-to-one when has only the zero solution. That requires a pivot in every variable column. Then two different inputs cannot have the same output.
Use a pivot in every row for onto
For an m-row matrix, the map is onto its m-dimensional output space when every row has a pivot. Then every target vector can be made from the columns.
This matrix takes two-component inputs to four-component outputs. The two columns are independent, so the map is one-to-one. It has at most two pivots, so it cannot reach every vector in the four-dimensional output space.
Require both conditions for invertibility
To be invertible as a map between its full input and output spaces, must have a pivot in every row and every column. That requires a square matrix of full rank. Its reduced row-echelon form is the identity matrix.