We are given an mxn matrix A whose cells contain positive numbers. We are interested in finding a path from A$[0][0]$ to A$[$m$-1][$n$-1]$ whose sum of values of the cells on the path is minimum. In going from the current cell to the next cell on the path we can only go to the cell on the immediate right $($when possible$)$ or the cell immediately below it $($when possible$).$ We are interested in a dynamic programming solution.

In a single sentence, define the matrix or array variable that your DP solution will fill in bottom up$.$ The definition's format should be

$>-$ what its value represents

In no more than three additional lines, describe the recurrence relation for computing this variable's values bottom up$.$ Including the initialization and corner cases.

Note that you are only describing how to compute the minimum path sum, not to actually find an optimal path itself, or even populate the data structure to assist with this process.