Definition: The inverse of a square matrix A is the matrix that multiplies with A to give the identity matrix; it exists only when the determinant is not zero.
What is an inverse matrix?
If A times B gives the identity matrix I (ones on the diagonal, zeros elsewhere), then B is the inverse of A, written A-1. For rows (a, b) and (c, d), swap a and d, negate b and c, and divide by the determinant ad – bc.
How does it solve a system?
Write the system as AX = B. Then X = A-1B.
Worked example: 3x + y = 7 and 5x + 2y = 12. The coefficient matrix has rows (3, 1) and (5, 2), determinant 6 – 5 = 1, and inverse with rows (2, -1) and (-5, 3). Multiply by the constants (7, 12): x = 14 – 12 = 2 and y = -35 + 36 = 1.
The solution is (2, 1), and 3(2) + 1 = 7 checks out.
What goes wrong?
Dividing by a zero determinant, or multiplying on the wrong side. Mathbot places the inverse on the left of the constants, since matrix order matters.
Where to learn it
- Solving Systems and Transforming the Plane – lesson
- Matrix Operations, Determinants and Inverses – lesson