# Evaluation of a Determinant

Evaluating the determinant of a matrix involves various methods, including expansion by minors, cofactor expansion, and using properties of determinants. The most common methods for evaluating determinants are:

### 1. Expansion by Minors or Cofactors:

• Method: Select a row or column, expand the determinant by minors or cofactors along that row or column.

• Formula: For a square matrix $A$ of order $n$:

$\text{det}\left(A\right)=\sum _{j=1}^{n}{a}_{ij}×{C}_{ij}\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\text{det}\left(A\right)=\sum _{j=1}^{n}{a}_{ij}×\left(-1{\right)}^{i+j}×{M}_{ij}$

where ${C}_{ij}$ is the cofactor of ${a}_{ij}$ and ${M}_{ij}$ is the minor of ${a}_{ij}$.

### 2. Row or Column Operations:

• Method: Apply elementary row or column operations to transform the matrix into a triangular or diagonal form.

• Formula: For an upper triangular or lower triangular matrix, the determinant is the product of the diagonal elements.

### 3. Properties of Determinants:

• Method: Utilize determinant properties to simplify the calculation or transform the matrix.

• Formula: Apply properties such as transposition, scaling rows or columns, or combining rows or columns to simplify the determinant calculation.

### Example:

Consider a $3×3$ matrix $A$:

$A=\left[\begin{array}{ccc}2& 3& 1\\ 5& 4& 6\\ 0& 7& 2\end{array}\right]$

To evaluate $\text{det}\left(A\right)$:

• One method is to expand along the first row:
$\text{det}\left(A\right)=2×\text{det}\left(\left[\begin{array}{cc}4& 6\\ 7& 2\end{array}\right]\right)-3×\text{det}\left(\left[\begin{array}{cc}5& 6\\ 0& 2\end{array}\right]\right)+1×\text{det}\left(\left[\begin{array}{cc}5& 4\\ 0& 7\end{array}\right]\right)$
• Another approach could be using row operations to convert $A$ into upper triangular form and then calculate the determinant as the product of the diagonal elements.

### Sarrus' Rule:

For a $3×3$ matrix:

$A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$

To find $\text{det}\left(A\right)$ using Sarrus' rule:

1. Extend the Matrix:
• Write the first two columns of the matrix to the right of the original matrix.
$\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]\begin{array}{cc}a& b\\ d& e\\ g& h\end{array}$
1. Multiply Diagonals:

• Multiply the elements along the three diagonals going from the upper left to the lower right.
2. Sum the Products:

• Add the products obtained in step 2 moving from left to right.
• Subtract the sum of the products obtained by moving from right to left.

### Example:

Consider the matrix:

$A=\left[\begin{array}{ccc}2& 3& 5\\ 1& -1& 4\\ 0& 2& -3\end{array}\right]$

Applying Sarrus' rule:

$\text{det}\left(A\right)=\left(2×\left(-1\right)×\left(-3\right)\right)+\left(3×4×0\right)+\left(5×1×2\right)-\left(0×\left(-1\right)×5\right)-\left(2×4×2\right)-\left(\left(-3\right)×1×2\right)$
$\text{det}\left(A\right)=\left(-6\right)+\left(0\right)+\left(10\right)-\left(0\right)-\left(16\right)-\left(-6\right)=-6+10-16+6=-6$

### Significance:

• Sarrus' rule simplifies the calculation of $3×3$ determinants by following a specific pattern without needing to compute minors or cofactors.

• This method is useful for relatively smaller matrices where applying direct computation by expanding determinants might be more time-consuming.