Ch 4 covers determinants of square matrices, their properties, cofactor expansion, adjoint and inverse of a matrix, and solving systems of linear equations using determinants (Cramer's rule).
A determinant is a scalar associated with a square matrix. For 2×2: |a b; c d| = ad−bc. Properties: |Aᵀ| = |A|, swapping rows/columns changes sign, proportional rows → determinant = 0, det(AB) = det(A)·det(B).
The cofactor of element aᵢⱼ is (−1)ⁱ⁺ʲ times the minor. Adjoint = transpose of cofactor matrix. A⁻¹ = adj(A)/|A| (exists only when |A| ≠ 0). AA⁻¹ = A⁻¹A = I.
For n equations in n unknowns: if D ≠ 0, unique solution exists. x = D₁/D, y = D₂/D, z = D₃/D where Dᵢ is obtained by replacing the ith column of coefficient matrix with constants.
Download: https://ncert.nic.in/textbook/pdf/lemh104.pdf | Complete book Part I: https://ncert.nic.in/textbook/pdf/lemh1ps.zip
A square matrix A has no inverse when its determinant is 0 (|A| = 0). Such a matrix is called singular. Geometrically, it maps space to a lower dimension, losing information that cannot be recovered.
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