Inverse of a 2×2 Matrix
Inverse of a 2×2 matrix (ad−bc ≠ 0).
Formula
Variables
Example
Inverse of [[1,2],[3,4]] = (1/−2)[[4,−2],[−3,1]] = [[−2,1],[1.5,−0.5]].
How it works
For a 2×2 matrix A = [[a,b],[c,d]], the inverse is the unique matrix that undoes its action, so multiplying A by A^{-1} gives the identity matrix. The formula comes from solving the four entries of the product A·A^{-1} so that the off-diagonal terms become 0 and the diagonal terms become 1; the factor 1/(ad−bc) appears because the determinant is the scale factor that must be nonzero for the system to have a solution. Geometrically, ad−bc tells you whether the linear transformation collapses the plane; if it is 0, the matrix squashes area to zero and cannot be reversed.
When to use it
Use this when you need to solve a 2-variable linear system, reverse a 2D linear transformation, or compute an explicit inverse in a proof or calculation. It is especially useful in linear algebra, analytic geometry, circuits, economics, and any setting where a 2×2 matrix models a linear relationship.
Worked example, step by step
Let A = [[2,5],[1,3]]. Step 1: Compute the determinant: ad−bc = (2)(3) − (5)(1) = 6 − 5 = 1. Step 2: Since the determinant is nonzero, the inverse exists. Step 3: Swap the diagonal entries and negate the off-diagonal entries: [[d,−b],[−c,a]] = [[3,−5],[−1,2]]. Step 4: Multiply by 1/(ad−bc) = 1/1 = 1. So A^{-1} = 1·[[3,−5],[−1,2]] = [[3,−5],[−1,2]]. Check: [[2,5],[1,3]]·[[3,−5],[−1,2]] = [[1,0],[0,1]], so the inverse is correct.
Common mistakes to avoid
- Forgetting to compute ad−bc first and trying to invert a matrix with determinant 0
- Swapping the wrong entries or only changing signs without also exchanging a and d
- Dropping the scalar 1/(ad−bc) or applying it to only part of the matrix
Did You Know?
Matrix inversion is the linear-algebra version of division — and impossible when the determinant is zero.
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