Using Inverses to Solve Systems

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Using Inverses to Solve Systems Rewrite a linear system as , then read off the unique solution . Solve the linear system for x and y using the inverse of the coefficient matrix: The system packaged as a single matrix equation — solving it means isolating the vector x. Nicely done. You solved a linear system by inverting its coefficient matrix and reading off . The method: rewrite , then — provided A is invertible. Invertibility check: guarantees a unique solution; here . The inverse: — swap the diagonal, negate the off-diagonal, divide by the determinant. The result: , i.e. , and substituting back gives exactly. When it pays off: once you have A^ -1 , every new right-hand side costs just one multiplication — handy when many systems share the same A . Treating a system as one matrix equation turns "solve for x and y " into a single product — the same idea scales to any system.

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