%5E-1%3Da-matrix-proof.jpeg "(a^-1)^-1=a Matrix Proof")
Matrix Proof: $(A^{-1})^{-1} = A$
Introduction
In linear algebra, the inverse of a matrix is a fundamental concept. It is used to solve systems of linear equations, find the determinant of a matrix, and perform various other operations. One important property of matrix inversion is that $(A^{-1})^{-1} = A$, which means that the inverse of the inverse of a matrix is the original matrix itself. In this article, we will provide a proof of this property using matrix algebra.
Preliminaries
Before we dive into the proof, let's recall some basic properties of matrix multiplication and inversion.
- Associativity of Matrix Multiplication: For any three matrices $A, B, C$ such that the multiplication is defined, we have $(AB)C = A(BC)$.
- Inverse of a Matrix: A matrix $A$ is said to have an inverse if there exists a matrix $B$ such that $AB = BA = I$, where $I$ is the identity matrix.
Proof
Let's assume that $A$ is a square matrix with an inverse, denoted by $A^{-1}$. We need to show that $(A^{-1})^{-1} = A$.
Step 1: Show that $(A^{-1})^{-1}$ exists
Since $A$ has an inverse, we know that $A^{-1}A = AA^{-1} = I$. This implies that $A^{-1}$ also has an inverse, which we will denote by $(A^{-1})^{-1}$.
Step 2: Show that $(A^{-1})^{-1}A = I$
Using the associativity of matrix multiplication, we have:
$(A^{-1})^{-1}A = (A^{-1})^{-1}AA^{-1} = ((A^{-1})^{-1}A)A^{-1} = IA^{-1} = A^{-1}$
Step 3: Show that $A(A^{-1})^{-1} = I$
Again using the associativity of matrix multiplication, we have:
$A(A^{-1})^{-1} = A(A^{-1})^{-1}A^{-1}A = A(A^{-1}(A^{-1})^{-1})A = AI = IA = A$
Step 4: Conclude that $(A^{-1})^{-1} = A$
From Steps 2 and 3, we have shown that $(A^{-1})^{-1}A = I$ and $A(A^{-1})^{-1} = I$. This implies that $(A^{-1})^{-1}$ is the inverse of $A$, and therefore, $(A^{-1})^{-1} = A$.
Conclusion
In this article, we have provided a proof that $(A^{-1})^{-1} = A$ for any square matrix $A$ with an inverse. This property is a fundamental result in linear algebra and has numerous applications in various fields, including physics, engineering, and computer science.
