3x3 Matrix Determinant

What is a Matrix Determinant?

The determinant of a matrix is a scalar value that can be computed from the elements of a square matrix. For a 3x3 matrix, the determinant provides important information about the matrix's properties and can be used in various mathematical and practical applications.

The Determinant Formula for a 3x3 Matrix

For a 3x3 matrix A:

\[A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\]

The determinant is calculated as:

\[det(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\]

Where:

• \(a, b, c, d, e, f, g, h, i\) are the elements of the matrix
• \(det(A)\) represents the determinant of matrix A

Calculation Steps

1. Multiply a by the determinant of its minor matrix (ei-fh)
2. Subtract b multiplied by the determinant of its minor matrix (di-fg)
3. Add c multiplied by the determinant of its minor matrix (dh-eg)
4. Sum these three terms to get the final determinant

Example Calculation

Let's calculate the determinant of matrix A:

\[A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}\]

Applying the formula:

\[det(A) = 1[(5 \times 9) - (6 \times 8)] - 2[(4 \times 9) - (6 \times 7)] + 3[(4 \times 8) - (5 \times 7)]\] \[det(A) = 1(45 - 48) - 2(36 - 42) + 3(32 - 35)\] \[det(A) = 1(-3) - 2(-6) + 3(-3)\] \[det(A) = -3 + 12 - 9 = 0\]

Visual Representation

This diagram visually represents the 3x3 matrix A used in our example calculation.