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Inverse matrices

Matrices

Japanese school year: University year 1

What you learn

This section covers the definition of inverse matrices, invertibility criteria via determinants, and explicit computational formulas. Inverse matrices are essential tools for solving matrix equations and performing coordinate transformations in geometry. Familiarity with matrix multiplication, identity matrices, and determinant calculations is required beforehand.

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Key points

This defines the inverse matrix A−1A^{-1}, which acts like a reciprocal for numbers. Multiplying AA by A−1A^{-1} from either side gives the identity matrix II, which behaves like the number 1.

AA−1=A−1A=IAA^{-1} = A^{-1}A = I

Use this formula to compute the inverse matrix A−1A^{-1} of a 2×22 \times 2 matrix. Swap diagonal entries aa and dd, change the signs of bb and cc, and divide the whole matrix by determinant ad−bcad - bc.

A−1=1ad−bc(d−b−ca)(ad−bc≠0)A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \quad (ad-bc \neq 0)

This formula gives the inverse of a matrix product ABAB. You multiply their individual inverses, but the order reverses so that B−1B^{-1} comes before A−1A^{-1}.

(AB)−1=B−1A−1(AB)^{-1} = B^{-1}A^{-1}

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