Q.State the condition used in the Hungarian Method to decide whether the current reduced matrix already gives an optimal assignment. Referring to Worked Example 4, state how many lines were needed to cover all zeros

The Hungarian Method's optimality test compares the minimum number of lines needed to cover every zero in the reduced matrix against the size of the matrix, $n$; only when the two are equal is the current set of zeros guaranteed to contain a full one-to-one assignment. …

The condition: after row and column reduction, find the minimum number of horizontal and vertical lines needed to cover every zero in the matrix. If this minimum equals $n$ (the size of the square matrix), the zeros contain a complete set of $n$ independent zeros and the current matrix is optimal. If the minimum is less than $n$, no full one-to-one assignment of zeros exists yet, and the matrix must be adjusted (Step 5 of the algorithm) before testing again.

Applied to Worked Example 4 ($n=3$):

(a) Immediately after row and column reduction, the zeros ($W_1J_3,W_2J_1,W_2J_2,W_3J_3$) could be covered using only row $W_2$ and column $J_3$ — 2 lines. Since $2<3$, the matrix was not yet optimal: no set of 3 independent zeros (one per row and column) existed at that stage. …

Believing that finding some set of zeros touching every row and column proves optimality — the actual test is the minimum number of lines needed to cover ALL zeros in t …