![[Formales]] ![[determinand_of_matrix__674206499.png]] [Matrix Determinant](https://www.ncl.ac.uk/webtemplate/ask-assets/external/maths-resources/core-mathematics/pure-maths/matrices/matrix-determinant.html#:~:text=Definition,or%20sometimes%20det(A)%20.)<br> In: *Newcastle University* > [!TLDR] > - A matrix's determinant is a scalar value derived from its elements. It can be thought of as a signature or a specific kind of summary for square matrices (i.e., matrices with an equal number of rows and columns). > - The determinant has several applications in linear algebra, including determining if a matrix is invertible. > - A matrix is invertible if and only if its determinant is non-zero. # Compute the Determinant ## 2x2 Matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ The determinant (denoted as $|A|$ or $\text{det}(A)$) is: $|A| = ad - bc$ ## 3x3 Matrix $A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$ The determinant is: $|A| = a(ei - fh) - b(di - fg) + c(dh - eg)$ # Larger matrices The determinant for larger matrices is usually computed with: - Laplace's expansion (cofactor expansion) - Reduction to triangular form (e.g., using the LU decomposition) - Using properties and operations that simplify the matrix (row operations). # Interpretations and Applications > [!multi-column] > >> [!NOTE] Volume Scaling >> For a $3 \times 3$ matrix, the determinant gives the volume scaling factor when the matrix is thought of as a linear transformation. For instance, if the determinant of a matrix is 3, it means that the matrix scales the volume of any shape it transforms by a factor of 3. > >> [!NOTE] Invertibility >> As mentioned, a matrix is invertible if and only if its determinant is non-zero. If the determinant is zero, the matrix is singular and does not have an inverse. > [!multi-column] > >> [!NOTE] Eigenvalues >> Determinants are crucial in finding the eigenvalues of a matrix, which have applications in various domains like differential equations, stability analysis, and more. > >> [!NOTE] Linear Dependence >> If the determinant of a matrix formed by vectors as its columns (or rows) is zero, then the vectors are linearly dependent. > [!multi-column] > >> [!NOTE] Area and Volume >> For $2 \times 2$ matrices, the determinant gives the area of the parallelogram spanned by its column (or row) vectors. >> For $3 \times 3$ matrices, it gives the volume of the parallelepiped spanned by its column (or row) vectors. > >> [!NOTE] Solution Existence >> In systems of linear equations, the determinant can be used to determine if the system has a unique solution.