Implement sparse matrix addition and multiplication

Implement sparse-matrix operations with explicit error handling.

Sparse matrix

A matrix A has dimensions r x c and is sparse, meaning most entries are zero.

You may choose a representation (e.g., hashmap/dictionary keyed by (row, col) for non-zero values, or compressed rows), but your implementation should be efficient for sparse inputs.

Tasks

Implement:

1. add(A, B) → returns A + B
2. mul(A, B) → returns A * B

Requirements

• If dimensions are incompatible:
  • For addition: A and B must have the same shape.
  • For multiplication: A is r x k , B must be k x c .
  • Incompatible dimensions must trigger clear error handling (e.g., raise an exception or return an error value, as specified by your interface).
• Preserve sparsity: avoid iterating over all r*c cells.

Input/Output (one reasonable interface)

• Input: two sparse matrices
• Output: a sparse matrix (or error)

Constraints (reasonable interview constraints)

• Up to 10^5 non-zero entries per matrix.
• Dimensions can be large (e.g., up to 10^9 ), so dense storage is not allowed.