The Augmented Matrix Method

Linear Algebra (Matrices) · Axiom Academy

LESSON The Augmented Matrix Method Find a matrix inverse by row-reduction: glue the identity onto A , reduce until the left side is I , and the right side becomes A^ -1 . Start with a square matrix A . Place the same-size identity matrix I to its right, separated by a bar, to form the augmented matrix . From now on every elementary row operation acts on the whole row — both halves at once. Left block: the matrix A we want to invert Right block: a fresh identity I , ready to record the operations 2. Reduce the Left to I — the Right Becomes A^ -1 Now row-reduce. Each operation is chosen to push the left block toward the identity. Because the operation hits the whole row, the right block changes in lockstep — and by the time the left side is I , the right side holds A^ -1 . Watch both halves move on every step. You can check it: multiplying A by this right-hand matrix returns the identity, AA^ -1 =I . 3. Why It Works — and When It Fails Each elementary row operation is the same as multiplying on the left by an elementary matrix E_k . If the operations turn A into I , then their product is exactly A^ -1 — and applying that same product to I builds A^ -1 on the right. That is the whole trick.

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