Math
Inverse matrices.
Definition
The inverse of a square matrix A, written A⁻¹, is the matrix such that A · A⁻¹ = A⁻¹ · A = I (the identity matrix). A matrix has an inverse only if its determinant is not zero.
How it works · 5 phases
Step by step.
- Calculate the determinant of the matrix.
- If the determinant is zero, the inverse does not exist.
- For a 2×2 matrix, swap the diagonal entries, negate the off-diagonal entries, and divide by the determinant.
- For larger matrices, use row reduction on [A | I] to get [I | A⁻¹].
- Verify by multiplying A · A⁻¹ to confirm you get the identity matrix.
Examples
Real-world.
- 1 Using A⁻¹ to solve the matrix equation AX = B as X = A⁻¹B
- 2 The inverse of [[2,1],[5,3]] is [[3,−1],[−5,2]]
- 3 Decoding encrypted messages using inverse matrix transformations
Key Fact
For a 2×2 matrix [[a,b],[c,d]]: A⁻¹ = (1/(ad−bc))·[[d,−b],[−c,a]]