### Determinants

**Concepts to remember:**

#### 1. Determinant of a matrix:

- Scalar value associated with a square matrix.
- It represents the scaling factor when the matrix is applied to a vector or geometric object.
- For a 2x2 matrix, the determinant is calculated as ad-bc, where a, b, c, and d are the elements of the matrix.

#### 2. Properties of determinants:

- Determinant of the product of two matrices equals the product of their determinants.
- If a matrix has a row or column of zeros, then its determinant is zero.
- Interchange of two rows or columns changes the sign of the determinant.
- Adding a multiple of one row (column) to another doesn’t alter the determinant.

#### 3. Minors and cofactors:

- Minors are determinants of submatrices formed by removing one row and one column from the main square matrix.
- Cofactors are defined as the product of the minor and the corresponding sign factor, depending on the row and column positions.

#### 4. Adjugate of a matrix:

- The adjoint (also called adjugate) of a square matrix is the transpose of its cofactor matrix.
- It is related to the inverse matrix and plays a significant role in determining matrix inverses.

#### 5. Inverse of a matrix using determinants:

- If a square matrix has a non-zero determinant, its inverse can be calculated by multiplying the adjoint matrix by the reciprocal of the determinant.

#### 6. Cramer’s rule for solving systems of linear equations:

- Cramer’s rule provides solutions for a system of linear equations with the same number of equations as variables.
- Each variable’s solution is calculated as the ratio of determinants, involving the coefficient matrix and matrices formed by replacing the corresponding variable’s column with the constant column.

#### 7. Area and volume using determinants:

- Determinants can be used to determine the area of quadrilaterals and volumes of parallelepipeds.
- For quadrilaterals, the two column vectors connecting the vertices form the matrix whose determinant represents twice the area.
- For parallelepipeds, three column vectors formed by the edges determine its volume through the determinant.

#### 8. Applications of determinants in geometry and physics:

- Determinants play a crucial role in various geometric transformations like rotations and reflections.
- In multivariable calculus, determinants are instrumental in calculating Jacobians and determining function extrema subject to constraints.
- They also have applications in physics, including electromagnetism and quantum mechanics, where determinant operators serve specific mathematical purposes.