Computing the Multiplicative Inverse of a 2×2 Matrix

Easy

In linear algebra, the multiplicative inverse (also known as the reciprocal matrix) of a square matrix A is another matrix denoted A⁻¹ that satisfies the fundamental property: $$A \times A^{-1} = A^{-1} \times A = I$$ where I is the identity matrix (a matrix with 1s on the main diagonal and 0s elsewhere). This property is analogous to how multiplying a number by its reciprocal yields 1. The 2×2 Case For a 2×2 matrix expressed as: $$A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$$ The inverse exists if and only if the determinant of the matrix is non-zero. The determinant for a 2×2 matrix is calculated as: $$\det(A) = ad - bc$$ When the determinant is non-zero, the inverse is given by the closed-form formula: $$A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}$$ Singular Matrices A matrix whose determinant equals zero is called a singular matrix or degenerate matrix. Such matrices have no multiplicative inverse because they represent transformations that "collapse" the space (losing a dimension), which cannot be reversed. Geometrically, a singular 2×2 matrix maps all points in a 2D plane onto a line or a single point. Your Task: Write a Python function that computes the multiplicative inverse of a given 2×2 matrix. The function should: Return a 2D list representing the inverse matrix when the determinant is non-zero Return None if the matrix is singular (i.e., when the determinant equals zero, making the matrix non-invertible)

Example 1

Input

matrix = [[4, 7], [2, 6]]

Output

[[0.6, -0.7], [-0.2, 0.4]]

Explanation

For the matrix [[a, b], [c, d]] = [[4, 7], [2, 6]]: **Step 1: Calculate the determinant** det = ad - bc = (4 × 6) - (7 × 2) = 24 - 14 = 10 **Step 2: Verify invertibility** Since det = 10 ≠ 0, the matrix is invertible. **Step 3: Apply the inverse formula** A⁻¹ = (1/det) × [[d, -b], [-c, a]] A⁻¹ = (1/10) × [[6, -7], [-2, 4]] A⁻¹ = [[0.6, -0.7], [-0.2, 0.4]] **Verification:** Multiplying the original matrix by its inverse yields the identity matrix: [[4, 7], [2, 6]] × [[0.6, -0.7], [-0.2, 0.4]] = [[1, 0], [0, 1]] ✓

Example 2

Input

matrix = [[1, 2], [2, 4]]

Output

None

Explanation

For the matrix [[a, b], [c, d]] = [[1, 2], [2, 4]]: **Step 1: Calculate the determinant** det = ad - bc = (1 × 4) - (2 × 2) = 4 - 4 = 0 **Step 2: Check invertibility** Since det = 0, the matrix is **singular** (non-invertible). The second row [2, 4] is exactly 2× the first row [1, 2], making the rows linearly dependent. This matrix collapses 2D space onto a line, which cannot be reversed. Therefore, the function returns **None**.

Example 3

Input

matrix = [[1, 0], [0, 1]]

Output

[[1.0, 0.0], [0.0, 1.0]]

Explanation

This is the **identity matrix** I₂. **Step 1: Calculate the determinant** det = ad - bc = (1 × 1) - (0 × 0) = 1 - 0 = 1 **Step 2: Apply the inverse formula** A⁻¹ = (1/1) × [[1, -0], [-0, 1]] = [[1, 0], [0, 1]] The identity matrix is its own inverse! This makes intuitive sense: the identity transformation leaves all vectors unchanged, so applying it twice (or any number of times) still leaves vectors unchanged. Mathematically: I × I = I.

Constraints

The input matrix will always be exactly 2×2 (two rows, two columns)
-10⁶ ≤ matrix[i][j] ≤ 10⁶
Matrix elements can be integers or floating-point numbers
When the inverse exists, round each element to 4 decimal places
A determinant is considered zero if its absolute value is less than 10⁻⁹

The 2×2 Case

For a 2×2 matrix expressed as:

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

The inverse exists if and only if the determinant of the matrix is non-zero. The determinant for a 2×2 matrix is calculated as:

$$\det(A) = ad - bc$$

When the determinant is non-zero, the inverse is given by the closed-form formula:

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

Singular Matrices

A matrix whose determinant equals zero is called a singular matrix or degenerate matrix. Such matrices have no multiplicative inverse because they represent transformations that "collapse" the space (losing a dimension), which cannot be reversed. Geometrically, a singular 2×2 matrix maps all points in a 2D plane onto a line or a single point.

Your Task: Write a Python function that computes the multiplicative inverse of a given 2×2 matrix. The function should:

Return a 2D list representing the inverse matrix when the determinant is non-zero

Return None if the matrix is singular (i.e., when the determinant equals zero, making the matrix non-invertible)