3×3 matrix determinant

3×3 Matrix Determinant

Determinant of a 3×3 Matrix

The determinant of a 3×3 matrix is a scalar value that can be computed from the elements of the matrix. The determinant of a 3×3 matrix is calculated by expanding along any row or column, typically the first row for simplicity. The formula involves multiplying each element by the determinant of the \(2 \times 2\) matrix that remains after removing the row and column of the element, and then applying appropriate signs.

For a given 3×3 matrix

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

the determinant, denoted as \(\text{det}(A)\) or \(|A|\), is calculated using the following formula:

\[
\text{det}(A) = a(ei – fh) – b(di – fg) + c(dh – eg)
\]

This is called the cofactor expansion along the first row, where each term corresponds to a minor of the matrix multiplied by the cofactor sign.

Example 1:

Given the matrix

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

To calculate the determinant, apply the formula:

\[
\text{det}(A) = 2\left(5 \times 9 – 6 \times 8\right) – 3\left(4 \times 9 – 6 \times 7\right) + 1\left(4 \times 8 – 5 \times 7\right)
\]

\[
\text{det}(A) = 2(45 – 48) – 3(36 – 42) + 1(32 – 35)
\]

\[
\text{det}(A) = 2(-3) – 3(-6) + 1(-3)
\]

\[
\text{det}(A) = -6 + 18 – 3 = 9
\]

So, the determinant of matrix \(A\) is \(9\).

Example 2:

Given the matrix

\[
B = \begin{pmatrix}
1 & 2 & 3 \\
0 & 4 & 5 \\
6 & 7 & 8
\end{pmatrix}
\]

Using the determinant formula

\[
\text{det}(B) = 1\left(4 \times 8 – 5 \times 7\right) – 2\left(0 \times 8 – 5 \times 6\right) + 3\left(0 \times 7 – 4 \times 6\right)
\]

\[
\text{det}(B) = 1(32 – 35) – 2(0 – 30) + 3(0 – 24)
\]

\[
\text{det}(B) = 1(-3) – 2(-30) + 3(-24)
\]

\[
\text{det}(B) = -3 + 60 – 72 = -15
\]

Thus, the determinant of matrix \(B\) is \(-15\).