Matrix Operations
| Operation | Requirement | Result size |
|---|---|---|
| Addition / Subtraction | Same dimensions | Same as input |
| Multiplication A×B | Cols(A) = Rows(B) | Rows(A)×Cols(B) |
| Transpose | Any matrix | Flipped dimensions |
| Determinant | Square matrix | Scalar value |
| Inverse | Square, det ≠ 0 | Same as input |
2×2 Key Formulas
Matrix A = [[a,b],[c,d]]
det(A) = ad − bc
A⁻¹ = (1/det) × [[d,−b],[−c,a]]
Aᵀ = [[a,c],[b,d]]
det(A) = ad − bc
A⁻¹ = (1/det) × [[d,−b],[−c,a]]
Aᵀ = [[a,c],[b,d]]
Frequently Asked Questions
When is matrix multiplication defined?
Matrix multiplication A×B is only defined when the number of columns in A equals the number of rows in B. An m×n matrix times an n×p matrix produces an m×p result. Note that matrix multiplication is generally not commutative: A×B ≠ B×A in most cases.
What does the determinant tell you?
The determinant is a scalar that encodes key properties of a square matrix. If det = 0, the matrix is singular (not invertible — the system has no unique solution). The absolute value of the determinant gives the scale factor for area/volume transformations. The sign indicates whether orientation is preserved.
What is an identity matrix?
The identity matrix I is the matrix equivalent of 1 — multiplying any matrix by I gives the same matrix back (A×I = I×A = A). For 2×2: [[1,0],[0,1]]. The inverse of A satisfies A×A⁻¹ = I.