4x4 Matrix Determinant

What is a Matrix Determinant?

The determinant of a square matrix is a scalar value that provides important information about the matrix's properties. For a 4x4 matrix, the determinant is a single number that represents a specific characteristic of the matrix.

The Determinant Formula

For a 4x4 matrix A, its determinant can be calculated using the Laplace expansion along the first row:

\[det(A) = a_{11}M_{11} - a_{12}M_{12} + a_{13}M_{13} - a_{14}M_{14}\]

Where:

• a_{ij} is the element in the i-th row and j-th column
• M_{ij} is the determinant of the 3x3 matrix formed by removing the i-th row and j-th column

Calculation Steps

1. Choose a row or column for expansion (usually the first row)
2. For each element in the chosen row/column:
   • Calculate the cofactor (the determinant of the 3x3 minor matrix)
   • Multiply the element by its cofactor, using alternating signs
3. Sum up all these products

Example Calculation

Let's calculate the determinant 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. Expand along the first row:

\[det(A) = 1M_{11} - 2M_{12} + 3M_{13} - 4M_{14}\]

2. Calculate each 3x3 determinant:

\[M_{11} = det\begin{bmatrix} 6 & 7 & 8 \\ 10 & 11 & 12 \\ 14 & 15 & 16 \end{bmatrix} = 0\] \[M_{12} = det\begin{bmatrix} 5 & 7 & 8 \\ 9 & 11 & 12 \\ 13 & 15 & 16 \end{bmatrix} = 0\] \[M_{13} = det\begin{bmatrix} 5 & 6 & 8 \\ 9 & 10 & 12 \\ 13 & 14 & 16 \end{bmatrix} = 0\] \[M_{14} = det\begin{bmatrix} 5 & 6 & 7 \\ 9 & 10 & 11 \\ 13 & 14 & 15 \end{bmatrix} = 0\]

3. Substitute and calculate:

\[det(A) = 1(0) - 2(0) + 3(0) - 4(0) = 0\]

Visual Representation

This diagram visually represents the 4x4 matrix A for which we calculated the determinant.