A matrix is invertible if it meets the following key conditions:
- It must be a square matrix (same number of rows and columns)
- Its determinant must be nonzero. A zero determinant means the matrix is singular (non-invertible)
- There exists another square matrix BBB such that when multiplied by the original matrix AAA, the product is the identity matrix III, i.e., AB=BA=IAB=BA=IAB=BA=I
- Equivalently, the matrix must have full rank, meaning its columns (and rows) are linearly independent
Intuitively, a nonzero determinant means the linear transformation represented by the matrix does not squash any dimension to zero, so it can be reversed
. In summary, a matrix is invertible if it is square and its determinant is not zero, ensuring the existence of a unique inverse matrix that yields the identity matrix upon multiplication
