# Determinants

### Determinants of a Matrix:

1. Definition:

• For a square matrix $A$, the determinant $\text{det}\left(A\right)$ is a scalar value calculated from its elements.
• It is denoted by vertical bars or $\text{det}\left(A\right)$ or $\mathrm{\mid }A\mathrm{\mid }$.
2. Calculation:

• For a $2×2$ matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, $\text{det}\left(A\right)=ad-bc$.
• For larger matrices, the calculation involves expansion by minors or other methods like row reduction.

For a 3x3 matrix $A$ of the form:

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

The determinant $\text{det}\left(A\right)$ is computed using the following formula:

$\text{det}\left(A\right)=aei+bfg+cdh-ceg-bdi-afh$

Expansion by Minors:

Another way to calculate the determinant of a $3×3$ matrix is by using expansion by minors. It involves breaking down the matrix into smaller determinants based on cofactors.

For matrix $A$:

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

The determinant $\text{det}\left(A\right)$ can be calculated as:

$\text{det}\left(A\right)=a\text{det}\left(\left[\begin{array}{cc}e& f\\ h& i\end{array}\right]\right)-b\text{det}\left(\left[\begin{array}{cc}d& f\\ g& i\end{array}\right]\right)+c\text{det}\left(\left[\begin{array}{cc}d& e\\ g& h\end{array}\right]\right)$ Where $\text{det}\left(\left[\begin{array}{cc}x& y\\ z& w\end{array}\right]\right)=xw-yz$.

### Properties of Determinants:

1. Multiplicative Property: $\text{det}\left(AB\right)=\text{det}\left(A\right)×\text{det}\left(B\right)$ for square matrices $A$ and $B$.

2. Transpose Property: $\text{det}\left(A\right)=\text{det}\left({A}^{T}\right)$ for any square matrix $A$.

3. Inverse Property: If $A$is invertible, then $\text{det}\left({A}^{-1}\right)=\frac{1}{\text{det}\left(A\right)}$.

4. Scalar Multiplication: $\text{det}\left(kA\right)={k}^{n}×\text{det}\left(A\right)$ for a square matrix $A$ of order $n$ and scalar $k$.

### Significance and Applications:

1. Matrix Inverses: A square matrix $A$ is invertible if and only if $\text{det}\left(A\right)\mathrm{\ne }0$.

2. System Solvability: For a system of linear equations $AX=B$, the system has a unique solution if and only if $\text{det}\left(A\right)\mathrm{\ne }0$.

3. Geometric Interpretation: The absolute value of the determinant represents the scaling factor in transformations and encodes information about the orientation change and volume scaling in space.

### Cramer's Rule:

Cramer's rule is a technique to solve systems of linear equations using determinants. For a system $AX=B$ where $A$ is a square matrix and $\text{det}\left(A\right)\mathrm{\ne }0$:

• The solution for ${x}_{i}$ can be expressed as ${x}_{i}=\frac{\text{det}\left({A}_{i}\right)}{\text{det}\left(A\right)}$ where ${A}_{i}$ is obtained by replacing the $i$th column of $A$ with $B$.

### Limitations:

• Calculating determinants becomes computationally expensive for larger matrices due to expansion by minors or other methods.