← MATH 01210 · Linear Algebra
MATH 01210 · Systems & matrices
Inverse matrices
Definition
A⁻¹ is the unique matrix with AA⁻¹=A⁻¹A=I, when it exists (A nonsingular / full rank / det≠0). Solve Ax=b via x=A⁻¹b when invertible.
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Find A⁻¹ by Gauss–Jordan on [A|I] → [I|A⁻¹], or use adjugate formula for 2×2/3×3.
2×2: [[a,b],[c,d]]⁻¹=(1/(ad−bc))[[d,−b],[−c,a]].
(AB)⁻¹=B⁻¹A⁻¹ when both invertible.
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Attempt each problem first, then open the worked solution.
1.Invert [[1,2],[3,5]] and solve Ax=(1,1).