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.