Proof For Properties of Invertible Matrix
There are various properties of invertible matrices, some of which are discussed as follows:
- (A-1)-1 = A
Proof:
If A is an invertible matrix then
AA-1 = I
Taking inverse on both sides
⇒ (AA-1)-1 = I-1
⇒ (A-1)-1A-1 = I [from theorem 2 (AB)-1 = B-1A-1]
Multiplying by A on both sides
(A-1)-1A-1A = IA
⇒ (A-1)-1I = A
⇒ (A-1)-1 = A
Hence, it is proved that (A-1)-1 = A
- (A1A2A3………..An)-1 = An-1An-1-1……….A2-1A1-1
Proof:
This can be proved by mathematical induction
for n = 2
(A1A2)-1 = A2-1A1-1 ……….(1)
This statement is true. [by theorem 2]
Let this is true for n = k
(A1A2A3……….Ak)-1 = Ak-1…………A2-1A1-1……..(2)
For n = k+1, you have to prove this.
(A1A2A3……….AkAk+1)-1
=((A1A2A3………Ak)Ak+1)-1
=((Ak-1…………A2-1A1-1)Ak+1)-1
=(Ak+1)-1 (Ak-1…………A2-1A1-1) [using theorem 2]
= Ak+1-1Ak-1………….A2-1A1-1
Hence, it is proved.
- AA-1= A-1A = In
Proof:
A matrix is invertible if AA-1 = I
Multiply by A on both sides
AAA-1 = AI
⇒ AI = A
Multiplying by A-1 on both sides
A-1AI = A-1A
⇒ I = A-1A
Hence, it is proved that AA-1 = I = A-1A
Read More,
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
Contact Us