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The detailed guide below is currently available in English.
What the determinant tells you
A determinant compresses a square matrix into one number. A nonzero determinant means the matrix is invertible and represents a transformation that preserves area (2×2) or volume (3×3) up to that scaling factor; a zero determinant means the matrix squashes space into fewer dimensions and has no inverse.
How the 3×3 inverse is built
Each entry of the inverse comes from the cofactors — the 2×2 determinants left when you strike out one row and column — with alternating signs. Transposing that cofactor grid produces the adjugate, and dividing by the determinant finishes the inverse. The 2×2 case is the familiar swap-and-negate shortcut over the determinant.
자주 묻는 질문
Why does my matrix have no inverse?
Its determinant is zero, so the matrix is singular — its rows (or columns) are linearly dependent and no matrix can undo the transformation it represents.
Why is A × B different from B × A?
Matrix multiplication is not commutative. Each result entry is a dot product of a row of the first matrix with a column of the second, so swapping the order generally changes the answer.