4x4 Inverse Matrix

What is a Matrix Inverse?

The inverse of a square matrix A, denoted as A^(-1), is a matrix that, when multiplied with A, results in the identity matrix. For a 4x4 matrix, if A * A^(-1) = A^(-1) * A = I_4, where I_4 is the 4x4 identity matrix, then A^(-1) is the inverse of A.

The Matrix Inverse Formula

For a 4x4 matrix A, its inverse A^(-1) is defined as:

\[A^{-1} = \frac{1}{det(A)} \cdot adj(A)\]

Where:

• det(A) is the determinant of A
• adj(A) is the adjugate matrix of A

Calculation Steps

1. Calculate the determinant of A
2. If the determinant is zero, the matrix is not invertible
3. If the determinant is non-zero, calculate the adjugate matrix of A
4. Divide each element of the adjugate matrix by the determinant

Example Calculation

Let's calculate the inverse of matrix A:

\[A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{bmatrix}\]

1. Calculate the determinant:

\[det(A) = 0\]

Since the determinant is zero, this matrix is not invertible.

For an invertible example, let's use:

\[A = \begin{bmatrix} 1 & 0 & 2 & -1 \\ 3 & 0 & 0 & 5 \\ 2 & 1 & 4 & -3 \\ 1 & 0 & 5 & 0 \end{bmatrix}\]

1. Calculate the determinant:

\[det(A) = 30\]

2. Calculate the adjugate matrix:

\[adj(A) = \begin{bmatrix} -20 & 5 & 10 & 0 \\ -10 & 0 & 10 & -5 \\ 0 & -5 & 0 & 5 \\ -5 & 6 & -5 & 0 \end{bmatrix}\]

3. Calculate the inverse:

\[A^{-1} = \frac{1}{30} \cdot \begin{bmatrix} -20 & 5 & 10 & 0 \\ -10 & 0 & 10 & -5 \\ 0 & -5 & 0 & 5 \\ -5 & 6 & -5 & 0 \end{bmatrix} = \begin{bmatrix} -2/3 & 1/6 & 1/3 & 0 \\ -1/3 & 0 & 1/3 & -1/6 \\ 0 & -1/6 & 0 & 1/6 \\ -1/6 & 1/5 & -1/6 & 0 \end{bmatrix}\]

Visual Representation

This diagram visually represents the original matrix A and its inverse A^(-1).