Inverse of a Square Matrix
Finding the matrix that "undoes" multiplication—the key to solving matrix equations.
Introduction
For numbers, the inverse of is because . For matrices, we seek an inverse matrix such that (the identity matrix). Not every matrix has an inverse—only nonsingular (invertible) matrices do.
Prerequisite Connection
You can multiply matrices and reduce to RREF using Gauss-Jordan elimination.
Today's Increment
We find the inverse of a square matrix using the augmentation method.
Why This Matters
Inverses solve matrix equations: . Cryptography relies on invertible matrices for encoding/decoding.
Key Concepts
Definition: Inverse Matrix
For square matrix , the inverse satisfies:
where is the identity matrix.
2×2 Inverse Formula
For :
Only works if (the determinant).
General Method: Gauss-Jordan
Form , reduce to RREF. If left side becomes , right side is :
Worked Examples
Example 1: 2×2 Formula (Basic)
Find the inverse of:
Step 1: Calculate determinant
Step 2: Apply formula
Answer:
Example 2: Gauss-Jordan Method (Intermediate)
Find the inverse using augmentation:
Step 1: Form
Step 2:
Step 3:
Answer:
Example 3: No Inverse (Advanced)
Find the inverse of:
Calculate determinant:
NO INVERSE EXISTS!
The determinant is 0, so is singular (non-invertible).
Common Pitfalls
Forgetting to check if determinant = 0
If , the inverse does NOT exist. Always check first!
Swapping positions wrong in 2×2 formula
Swap and (on diagonal), negate and (off-diagonal).
Not verifying with
Always multiply your answer by the original to confirm you get the identity matrix.
Real-World Application
Cryptography: Hill Cipher
The Hill cipher uses an invertible matrix to encrypt messages. Each letter becomes a number, grouped into vectors, and multiplied by a secret key matrix. To decrypt, multiply by the inverse of the key matrix. If someone chooses a singular matrix as their key, decryption is impossible!
Encrypt: | Decrypt:
Practice Quiz
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