Determinant of Inverse Matrix
To understand the determinant of the inverse matrix, let’s first define what the inverse of a matrix is.
The inverse of a square matrix A, denoted as A−1, is a matrix such that when it’s multiplied by A, the result is the identity matrix I. Mathematically, if A⋅A−1 = I, then A−1 is the inverse of A.
Now, the determinant of the inverse matrix, denoted as det(A−1), is related to the determinant of the original matrix A. Specifically, it can be expressed by the formula:
det(A−1) = 1/det(A)
This formula illustrates an important relationship between the determinants of a matrix and its inverse. If the determinant of A is non-zero, meaning det(A)≠0, then the inverse matrix exists, and its determinant is the reciprocal of the determinant of A. Conversely, if det(A)=0, the matrix A is said to be singular, and it does not have an inverse.
Here are some key points about the determinant of the inverse matrix:
- Non-Singular Matrices: For non-singular matrices (those with non-zero determinants), their inverses exist, and the determinant of the inverse is the reciprocal of the determinant of the original matrix.
- Singular Matrices: Singular matrices (those with zero determinants) do not have inverses. Attempting to find the inverse of a singular matrix results in an undefined or non-existent inverse.
- Geometric Interpretation: The determinant of a matrix measures how it scales the space. Similarly, the determinant of the inverse matrix measures the scaling effect of the inverse transformation. If the original transformation expands the space, its inverse contraction will be inversely proportional, and vice versa.
Determinant of a Matrix with Solved Examples
Determinant of a Matrix is defined as the function that gives the unique output (real number) for every input value of the matrix. Determinant of the matrix is considered the scaling factor that is used for the transformation of a matrix. It is useful for finding the solution of a system of linear equations, the inverse of the square matrix, and others. The determinant of only square matrices exists.
Table of Content
- Determinant of Matrix Calculator
- Definition of Determinant of Matrix
- Determinant of a 1×1 Matrix
- Determinant of 2×2 Matrix
- Determinant of a 3×3 Matrix
- Determinant of 4×4 Matrix
- Determinant of Identity Matrix
- Determinant of Symmetric Matrix
- Determinant of Skew-Symmetric Matrix
- Determinant of Inverse Matrix
- Determinant of Orthogonal Matrix
- Physical Significance of Determinant
- Laplace Formula for Determinant
- Properties of Determinants of Matrix