Determinant Matrix and Properties – Explained

The determinant is a fundamental concept in linear algebra that provides important information about a square matrix.

It is denoted by $|A|$ or $\det(A)$, where $A$ is the matrix. The determinant is only defined for square matrices.

 

Determinant of a $2\times 2$ Matrix

Let’s start with a $2\times 2$ matrix:
\[
A = \begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\]

The determinant of $A$, denoted as $|A|$ or $\det(A)$, is calculated using the formula:
\[
\det(A) = ad – bc
\]

So, the determinant of a $2\times 2$ matrix is simply the product of the elements on the main diagonal (from top-left to bottom-right) minus the product of the elements on the other diagonal.

 

Determinant of a $3\times 3$ Matrix

For a $3\times 3$ matrix:
\[
A = \begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{bmatrix}
\]

The determinant of $A$ is calculated as follows:
\[
\det(A) = aei + bfg + cdh – ceg – afh – bdi
\]

To find the determinant of a $3\times 3$ matrix, you need to calculate the sum of the products of the three diagonals parallel to the main diagonal and subtract the sum of the products of the three diagonals parallel to the other diagonal.

 

Determinant of a Larger Matrix

For larger square matrices, the calculation of the determinant becomes more complex. One common method is to use cofactor expansion.

Let $A$ be an $n\times n$ matrix. The determinant of $A$ can be calculated by expanding along any row or column. For simplicity, let’s assume we expand along the first row:

\[
\det(A) = a_{11}C_{11} – a_{12}C_{12} + a_{13}C_{13} – \ldots + (-1)^{1+n}a_{1n}C_{1n}
\]

In this formula, $a_{ij}$ represents the element in the $i$th row and $j$th column of matrix $A$. $C_{ij}$ is the cofactor of $a_{ij}$, which is calculated as the determinant of the submatrix obtained by removing the $i$th row and $j$th column.

This expansion continues recursively until we reach a $2\times 2$ submatrix, as explained earlier.

 

Properties of the Determinant

The determinant of a matrix has several important properties:

1. If any two rows (or columns) of a matrix are interchanged, the determinant changes its sign.
2. If all the elements of a row (or column) are multiplied by a scalar $k$, the determinant is multiplied by $k$.
3. If two rows (or columns) of a matrix are proportional, i.e., one row (or column) is a scalar multiple of another, the determinant is zero.
4. The determinant of the product of two matrices is equal to the product of their determinants: $\det(AB) = \det(A) \cdot \det(B)$.

These properties make the determinant a powerful tool in linear algebra and have applications in areas such as solving systems of linear equations and computing matrix inverses.