gen-define-the-determinant-and-give-one-short-worked-example-88d9

Define the determinant and give one short worked example.

# The Determinant ## Definition The **determinant** is a scalar value associated with a square matrix that encodes certain properties of the linear transformation it represents. For an n×n matrix A, we denote it as det(A) or |A|. **For a 2×2 matrix:** $$\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$$ **For larger matrices**, determinants are computed using cofactor expansion or other methods. ## Worked Example Find the determinant of $A = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$ **Solution:** $$\det(A) = (3)(4) - (1)(2) = 12 - 2 = 10$$ Therefore, det(A) = 10. --- **Quick interpretation:** Since det(A) ≠ 0, this matrix is invertible, and the linear transformation it represents scales areas by a factor of 10.