Inverse matrices

Math

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

  1. Calculate the determinant of the matrix.
  2. If the determinant is zero, the inverse does not exist.
  3. For a 2×2 matrix, swap the diagonal entries, negate the off-diagonal entries, and divide by the determinant.
  4. For larger matrices, use row reduction on [A | I] to get [I | A⁻¹].
  5. Verify by multiplying A · A⁻¹ to confirm you get the identity matrix.

Examples

  • Using A⁻¹ to solve the matrix equation AX = B as X = A⁻¹B
  • The inverse of [[2,1],[5,3]] is [[3,−1],[−5,2]]
  • 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]]

Related Terms

Matrix operations Determinants Solving systems with matrices

Study This Concept

Practice inverse matrices with free review games in these units:

Pre-Calculus Unit 7: Systems and Matrices