Matrix Inverses

Algebra 2 · Axiom Academy

Some matrices can be undone. Meet A^ -1 — the matrix with AA^ -1 =I — and the quick formula that builds it for any 2×2. Multiplying a matrix by its inverse undoes it. Formally, A^ -1 is the matrix for which AA^ -1 = A^ -1 A = I , where I is the identity matrix. The identity I is the matrix equivalent of 1 — it has 1's on the diagonal and 0's everywhere else: For any 2×2 matrix there's a direct formula. Start with Three moves build the inverse: swap the diagonal, negate the off-diagonal, and divide by the determinant . 3. Worked Example: Finding an Inverse Everything hinges on the determinant — the main-diagonal product minus the anti-diagonal product. Watch it get computed, then the inverse falls right out. 4. Solving a System with the Inverse The real payoff is solving systems. Write the system as one matrix equation AX = B , then multiply both sides on the left by A^ -1 . Because A^ -1 A = I , the A cancels and X stands alone: Reusing A^ -1 from the previous example: So x = 2 and y = 1 — one matrix multiplication solved both equations at once. A matrix has no inverse exactly when its determinant is zero. These are the singular matrices — the in the formula would divide by zero. Since , this matrix has no inverse . Quick check: always compute the determinant first. If it is 0, stop — there is no inverse. You've seen what a matrix inverse is, how to build one for a 2×2, and how it cracks a whole system of equations in a single step. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Algebra 2 course.