In mathematics, the determinant is a scalar value that is a function of the entries of a Square matrix. It allows characterizing some properties of the Matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is [an Invertible matrix] and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix A is denoted \(\determinant{A}\), \(\operatorname{det} A\), or \(|A|\).
\(\determinant{A_{2 \times 2}}\)
\[\determinant{\begin{bmatrix}a & b \\ c & d\end{bmatrix}} = ad - bc\]
\(\determinant{A_{3 \times 3}}\)
\[\determinant{\begin{bmatrix}a & b &c \\ d & e & f \\ g & h & i\end{bmatrix}} = a \determinant{\begin{bmatrix}e & f \\ h & i\end{bmatrix}} - b \determinant{\begin{bmatrix}d & f \\ g & i\end{bmatrix}} + c \determinant{\begin{bmatrix}d & e \\ g & h\end{bmatrix}}\]
Each determinant of a \(2 \times 2\) matrix in this equation is called a minor of the matrix \(A\). This procedure can be extended to give a recursive definition for the determinant of an \(n \times n\) matrix, known as Laplace expansion.
Geometric meaning
The absolute value of \(\determinant{A}\) represents the factor by which \(A\) scales areas.
Properties
- The determinant of a matrix composed of linearly dependent row-vectors or column vectors is 0
- \(\operatorname{det}(I) = 1\)
- \(\operatorname{det}(\begin{bmatrix}a & . & .\\ b & . & . \\ c & . & . \end{bmatrix}) = x\), \(\operatorname{det}(\begin{bmatrix}ra & . & .\\ rb & . & . \\ rc & . & . \end{bmatrix}) = rx\)