Matrix Inverse Calculator (2×2) Formula
Understand the math behind the matrix inverse calculator (2×2). Each variable explained with a worked example.
Formulas Used
Determinant
determinant = detInv A
inv_a = det != 0 ? d / det : 0Inv B
inv_b = det != 0 ? -b / det : 0Inv C
inv_c = det != 0 ? -c / det : 0Inv D
inv_d = det != 0 ? a / det : 0Variables
| Variable | Description | Default |
|---|---|---|
a | a (row 1, col 1) | 4 |
b | b (row 1, col 2) | 7 |
c | c (row 2, col 1) | 2 |
d | d (row 2, col 2) | 6 |
det | Derived value= a * d - b * c | calculated |
How It Works
2×2 Matrix Inverse
Formula
For matrix A = [[a, b], [c, d]] with det ≠ 0:
A⁻¹ = (1/det) × [[d, -b], [-c, a]]
where det = ad - bc.
Steps
1. Calculate the determinant 2. Swap a and d 3. Negate b and c 4. Divide every element by the determinant
Verification
A × A⁻¹ = I (identity matrix)
Worked Example
Find the inverse of [[4, 7], [2, 6]].
- 01det = 4×6 - 7×2 = 24 - 14 = 10
- 02Inverse = (1/10) × [[6, -7], [-2, 4]]
- 03= [[0.6, -0.7], [-0.2, 0.4]]
Frequently Asked Questions
What is a matrix inverse?
The inverse of matrix A is the matrix A⁻¹ such that A × A⁻¹ = I (identity). It "undoes" the transformation represented by A.
When does a matrix have no inverse?
A matrix has no inverse when its determinant is zero. Such matrices are called singular.
Why is the matrix inverse useful?
Matrix inverses are used to solve systems of linear equations (x = A⁻¹b), in computer graphics for inverse transformations, and in statistics for regression analysis.
Learn More
Guide
Introduction to Matrices - Complete Guide
Learn the fundamentals of matrices including notation, operations, determinants, inverses, and applications in systems of equations, transformations, and data science.
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