Matrix Calculator – Operations Decision Framework
Quickly pick the correct matrix operation, verify dimension rules, and sanity‑check results. No long lecture—just what you need.
1. Operation Summary
- A + B: same (m×n)
- A − B: same (m×n)
- A × B: (m×k)(k×n) → (m×n)
- Scalar · A: multiply each entry
- det(A): square only
- A⁻¹: square + det(A) ≠ 0
- Aᵀ: (m×n) → (n×m)
2. Decision Steps
- Goal? Combine matrices, transform, or solve system.
- Check shapes: Write dimension pair quickly (e.g., (2×3)(3×2)).
- Square? Needed for det / inverse.
- Determinant zero? If 0, inverse not possible.
- Interpret result: Dimension must match expectation.
3. Guardrails
- Dimension mismatch: most common error—always jot shapes first.
- Inverse amplification: large condition numbers amplify rounding (not shown here, but watch huge swings).
- Det close to 0 → numerical instability.
4. Quick Reference
- (AB)ᵢⱼ = Σ AᵢₖBₖⱼ
- 2×2 det = ad − bc
- (Aᵀ)ᵀ = A
- (AB)ᵀ = BᵀAᵀ
- A A⁻¹ = I
- I * A = A
5. Sanity Checks
- Add/Subtract: Result entries near average magnitude of inputs.
- Multiply: (m×n)(n×p) result shape (m×p) correct?
- Inverse: A · A⁻¹ visually near identity (diagonal ≈1, off‑diag ≈0).
6. Pitfalls
- Ordering matters: AB ≠ BA generally.
- Forgetting to force square for inverse/det (tool auto-enforces).
- Misreading transpose: it's not element‑wise function application.
7. FAQ
Why NaN? Empty/invalid cell or illegal dimension pairing.
Inverse failing? Determinant is zero (singular matrix).
Large weird numbers? Possible near‑singular matrix causing amplification.
8. Action Tip
Write dimension strings BEFORE pressing calculate—this single habit prevents most mistakes.