# Determinants

## Determinants Basics

- A
**determinant**is a special numerical value calculated only for square matrices. - It can be computed by applying a series of transformations and operations to the rows and/or columns of the matrix.
- The determinant of a matrix gives us important information about the matrix, particularly in equations and systems of linear equations that the matrix represents.

## Calculation of Determinants

- The determinant of a
**1x1 matrix**is the single value in the matrix. - The determinant of a
**2x2 matrix**is calculated using the rule`ad - bc`

, where`a`

,`b`

,`c`

and`d`

are the elements of the matrix. - For a
**3x3 matrix**, you can use the rule of**Sarrus**: repeat the first two columns of the matrix to the right of the matrix, then multiply the elements of the down-right diagonals, sum them up and subtract the sum of the products of the up-right diagonals. - For matrices larger than 3x3, the determinant can be calculated by the method of
**cofactors**,**minors**and the**expansion of a row or column**.

## Properties of Determinants

- The determinant of the identity matrix is
**1**. - If we
**transpose**a matrix (swap rows and columns), the determinant of the new matrix is the same as the determinant of the original. - If any row or column in a square matrix is multiplied by a scalar, then the determinant of the new matrix is the determinant of the original matrix multiplied by the scalar.
- If a square matrix has a
**row or column of zeroes**, its determinant is**0**. - If two rows or columns of a matrix are identical, the determinant is
**0**. - The determinant of the product of two square matrices is the
**product of their determinants**, i.e.,`Det(AB) = Det(A) * Det(B)`

. - If the determinant of a matrix is
**0**, the matrix is said to be**singular**, and it has no**inverse**.

## Applications of Determinants

- Determinants are used to calculate the
**inverse of a matrix**. - They are used to solve
**systems of linear equations (Cramer’s Rule)**. - They can be used to check the
**consistency of the system of equations**. - They also play a crucial role in various operations in
**3D geometry**.