Lesson 3: The Inverse of a Matrix
Estimated time: 35-45 minutes
Learning Objectives
- Define the inverse of a square matrix
- Compute the inverse of a 2x2 matrix using the formula
- Compute the inverse of any square matrix using the [A | I] row reduction method
- Use inverses to solve systems of equations: x = A^{-1}b
- State properties of inverses, including (AB)^{-1} = B^{-1}A^{-1}
What Is an Inverse?
Inverse Matrix: An n x n matrix A is invertible (or nonsingular) if there exists an n x n matrix A^{-1} such that AA^{-1} = A^{-1}A = I_n. The matrix A^{-1} is called the inverse of A.
Not every matrix has an inverse. A matrix that is not invertible is called singular.
The 2x2 Inverse Formula
2x2 Inverse Formula: If A = [ a b ; c d ] and ad - bc is not zero, then:
The quantity ad - bc is the determinant of A (studied in Module 3). If ad - bc = 0, A is singular.
Worked Example
Find the inverse of A = [ 2 1 ; 5 3 ].
Step 1: Compute ad - bc = 2(3) - 1(5) = 6 - 5 = 1.
Step 2: Apply the formula: A^{-1} = (1/1) [ 3 -1 ; -5 2 ] = [ 3 -1 ; -5 2 ].
Verify: AA^{-1} = [ 2(3)+1(-5), 2(-1)+1(2) ; 5(3)+3(-5), 5(-1)+3(2) ] = [ 1 0 ; 0 1 ] = I. ✓
The Row Reduction Method for Larger Matrices
For matrices larger than 2x2, use this algorithm:
Algorithm: Finding A^{-1}
- Form the augmented matrix [ A | I ].
- Row reduce using Gauss-Jordan elimination.
- If the left side reduces to I, the right side is A^{-1}: [ I | A^{-1} ].
- If you get a row of zeros on the left side, A is not invertible.
Worked Example: 3x3 Inverse
Find the inverse of A = [ 1 0 1 ; 0 1 0 ; 1 0 0 ].
Form [ A | I ]:
[ 0 1 0 | 0 1 0 ]
[ 1 0 0 | 0 0 1 ]
R_3 - R_1 → R_3:
[ 0 1 0 | 0 1 0 ]
[ 0 0 -1 | -1 0 1 ]
(-1)R_3 → R_3, then R_1 - R_3 → R_1:
[ 0 1 0 | 0 1 0 ]
[ 0 0 1 | 1 0 -1 ]
A^{-1} = [ 0 0 1 ; 0 1 0 ; 1 0 -1 ].
Solving Systems with Inverses
If A is invertible, the system Ax = b has the unique solution:
Example
Solve [ 2 1 ; 5 3 ] x = [ 4 ; 11 ].
We found A^{-1} = [ 3 -1 ; -5 2 ]. So x = A^{-1}b = [ 3(4)+(-1)(11) ; (-5)(4)+2(11) ] = [ 12-11 ; -20+22 ] = [ 1 ; 2 ].
Properties of Inverses
Key Properties (assuming all matrices are invertible and square):
- (A^{-1})^{-1} = A
- (AB)^{-1} = B^{-1} A^{-1} (order reverses, like transpose!)
- (A^T)^{-1} = (A^{-1})^T
- (cA)^{-1} = (1/c) A^{-1} for nonzero scalar c
Check Your Understanding
1. Find the inverse of A = [ 3 2 ; 1 1 ] using the 2x2 formula.
2. Is the matrix [ 2 4 ; 1 2 ] invertible?
3. If A and B are invertible 3x3 matrices, what is (AB)^{-1}?
4. Solve [ 1 2 ; 3 7 ] x = [ 1 ; 1 ] using the inverse.
Key Takeaways
- A^{-1} exists only for square matrices with nonzero determinant
- 2x2 formula: swap diagonals, negate off-diagonals, divide by det
- General method: row reduce [ A | I ] to [ I | A^{-1} ]
- Solving systems: if A is invertible, x = A^{-1}b
- (AB)^{-1} = B^{-1}A^{-1} -- order reverses