Invertible Matrix Determinant
We define that for any square matrix A the determinant of the inverse of the square matrix A is the reciprocal of the determinant of the square matrix, i.e.
Det (A-1) = 1/Det(A)
Proof of Invertible Matrix Determinant
The Proof for Det (A-1) = 1/Det(A) is discussed below,
We know that,
Det(A × B) = Det (A) × Det(B)
A × A-1 = In (property of invertible matrix)
⇒ Det(A ×A-1) = Det(In)
⇒ Det(A) × Det(A-1) = Det(In) {Det(In) = 1}
⇒ Det(A) × Det(A-1) = 1
⇒ Det(A-1) = 1 / Det(A)
Hence, proved.
Invertible Matrix
Invertible matrices are defined as the matrix whose inverse exists. We define a matrix as the arrangement of data in rows and columns, if any matrix has m rows and n columns then the order of the matrix is m × n where m and n represent the number of rows and columns respectively.
We define invertible matrices as square matrices whose inverse exists. They are non-singular matrices as their determinant exists. There are various methods to calculate the inverse of the matrix.
In this article, we will learn about, What are Invertible Matrices? Invertible Matrices Examples, Invertible Matrix Theorems, Invertible Matrix Determinant, and others in detail.
Table of Content
- What is Invertible Matrix?
- Invertible Matrix Example
- Matrix Inversion Methods
- Invertible Matrix Theorem
- Invertible Matrix Properties
- Invertible Matrix Determinant