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2×2 determinants

Matrices

Japanese school year: University year 1

What you learn

This section covers the definition of the 2x2 determinant, its geometric interpretation as signed area, and its role in testing matrix invertibility. Determinants are crucial for determining whether inverse matrices exist and solving linear systems. Prior understanding of matrix notation and basic plane geometry is recommended.

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Key points

This fundamental formula calculates the determinant of a 2×22 \times 2 matrix. Subtract the cross-diagonal product bcbc from the main diagonal product adad; if this value is not 0, the matrix has an inverse.

∣abcd∣=ad−bc\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

This formula calculates the area SS of the parallelogram formed by two vectors (a,c)(a, c) and (b,d)(b, d). Since the determinant ad−bcad - bc can be negative, taking its absolute value gives the true positive area.

S=∣ad−bc∣S = |ad - bc|

This property shows that the determinant of a product ABAB equals the product of individual determinants det⁡A\det A and det⁡B\det B. It lets you find the determinant of ABAB without performing the matrix multiplication first.

det⁡(AB)=(det⁡A)(det⁡B)\det(AB) = (\det A)(\det B)

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