2x2 Inverse Matrix

What is a 2x2 Inverse Matrix?

The inverse of a 2x2 matrix A, denoted as A⁻¹, is a matrix that when multiplied with A, results in the identity matrix. In other words, AA⁻¹ = A⁻¹A = I, where I is the 2x2 identity matrix.

The Formula

For a 2x2 matrix A:

\[A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\]

Its inverse A⁻¹ is calculated as:

\[A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\]

Where:

• |A| = ad - bc is the determinant of A
• The matrix [d -b; -c a] is the adjugate of A

Calculation Steps

1. Calculate the determinant |A| = ad - bc
2. If the determinant is zero, the matrix is not invertible
3. If the determinant is non-zero, calculate the adjugate matrix
4. Divide the adjugate matrix by the determinant to get the inverse

Example

Let's find the inverse of this matrix:

\[A = \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix}\]

Step-by-step calculation:

1. Determinant: |A| = (4 × 6) - (7 × 2) = 24 - 14 = 10
2. Adjugate matrix: \[adj(A) = \begin{bmatrix} 6 & -7 \\ -2 & 4 \end{bmatrix}\]
3. Inverse: \[A^{-1} = \frac{1}{10} \begin{bmatrix} 6 & -7 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{bmatrix}\]

Visual Representation

This diagram illustrates the original matrix A and its inverse A⁻¹.