How Do I Find the Inverse of a 2x2 Matrix?

For [[a,b],[c,d]], the inverse is (1/(ad-bc)) × [[d,-b],[-c,a]]. Example: [[1,2],[3,4]] has det = 1×4 - 2×3 = -2, so the inverse is [[-2, 1],[1.5, -0.5]].

The Formula

Given a 2×2 matrix:

A = | a  b |
    | c  d |

A^(-1) = (1 / det(A)) * |  d  -b |
                         | -c   a |

where det(A) = ad - bc

The matrix is invertible only when det(A) is not zero.

Worked Example

Find the inverse of:

A = | 1  2 |
    | 3  4 |
  1. Compute the determinant: det = 1×4 - 2×3 = 4 - 6 = -2
  2. Swap a and d, negate b and c: 
    |  4  -2 |
| -3   1 |
  3. Multiply by 1/det = 1/(-2) = -0.5: 
    | -2.0   1.0 |
|  1.5  -0.5 |

Verification

Multiply A × A^(-1) to confirm:

| 1  2 |   | -2.0   1.0 |   | 1*(-2)+2*(1.5)    1*(1)+2*(-0.5)  |   | 1  0 |
| 3  4 | * |  1.5  -0.5 | = | 3*(-2)+4*(1.5)    3*(1)+4*(-0.5)  | = | 0  1 |

The result is the identity matrix, confirming the inverse is correct.

When Does No Inverse Exist?

A matrix is singular (non-invertible) when its determinant equals zero. Example: [[2,4],[1,2]] has det = 2×2 - 4×1 = 0. This means the rows are linearly dependent (row 1 = 2 × row 2), so no inverse exists.

Use the ML3X matrix calculator to compute inverses for matrices up to 5×5 with full step-by-step solutions.

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