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Determinants and adjoint

1️⃣ Determinants A determinant is a special numerical value associated with a square matrix (a matrix having equal number of rows and columns). It is denoted by |A| or det(A). The determinant helps to: Check whether a matrix is invertible or not Solve system of linear equations (Cramer’s Rule) Find area and volume in geometry Study properties of linear transformations --- 🔹 Determinant of a 2×2 Matrix If A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} then |A| = ad - bc --- 🔹 Determinant of a 3×3 Matrix For a 3×3 matrix, the determinant can be calculated by: Cofactor expansion (Laplace expansion) Sarrus Rule --- 2️⃣ Adjoint (Adjugate Matrix) The adjoint of a square matrix A is defined as the transpose of its cofactor matrix. It is denoted by: adj(A) --- 🔹 Steps to Find Adjoint 1. Find the minor of each element. 2. Find the cofactor of each element. 3. Form the cofactor matrix. 4. Take the transpose of the cofactor matrix. The resulting matrix is called the adjoint of A. --- 🔹 Important Formula A^{-1} = \frac{1}{|A|} \, adj(A) This formula is valid only when: |A|
eq 0 --- 🔹 Key Points If |A| = 0 → The matrix is singular (no inverse exists). If |A| ≠ 0 → The matrix is non-singular (inverse exists). Adjoint is mainly used to find the inverse of a matrix. If you want, I can also prepare a detailed example with full solution.