Introduction

In the realm of linear algebra, the determinant is a special number that can be calculated from a square matrix. It encodes certain properties of the linear map represented by the matrix, providing a compact way to understand the matrix's behavior. The determinant of a matrix A is often denoted as |A| or det(A).

Applications

Role in Linear Systems

The determinant has a key role in the solution of linear systems. For a system of equations defined by AX = B, if the determinant of matrix A (det(A)) is non-zero, the system has a unique solution. Conversely, if det(A) is zero, the system either has no solutions or an infinite number of solutions.

Matrix Inversibility

The determinant also informs us about the invertibility of a matrix. A square matrix is invertible if and only if its determinant is non-zero.

Volume and Orientation

In geometric terms, the absolute value of the determinant of a matrix gives the scale factor by which area, volume, etc., are multiplied under the associated linear transformation. The sign of the determinant reflects whether the transformation preserves orientation.

References

• Determinant - Wikipedia
• Determinant - MathWorld