Difference between revisions of "Invertible Matrices"
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<math>B^{-1}=A</math> and <math>A</math> can also be called an inverse of <math>B</math> | <math>B^{-1}=A</math> and <math>A</math> can also be called an inverse of <math>B</math> | ||
== Invertible Matrices Theorem == | |||
=== Theorem 1 === | |||
If there exists an inverse of a square matrix, it is always unique. | |||
'''Proof:''' | |||
Let <math>A</math> is a square matrix of order <math>n \times n</math>. Let matrices <math>B</math> and <math>C</math> to be inverses of matrix <math>A</math>. | |||
Now <math>AB=BA=I</math> since <math>B</math> is the inverse of matrix <math>A</math>. | |||
Similarly, <math>AC=CA=I</math>. | |||
But <math>B=BI=B(AC)=(BA)C=IC=C</math> | |||
This proves <math>B=C</math> or <math>B</math> and <math>C</math> are the same matrices. | |||
=== Theorem 2 === | |||
If <math>A</math> and <math>B</math> are matrices of the same order and are invertible, then <math>(AB)^{-1}=B^{-1}A^{-1}</math> | |||
=== Proof: === | |||
As per the definition of inverse of a matrix | |||
<math>(AB)(AB)^{-1}=I</math> | |||
<math>A^{-1}(AB)(AB)^{-1}=A^{-1}I</math> --------- Multiply by <math>A^{-1}</math> on both sides | |||
<math>(A^{-1}A)B(AB)^{-1}=A^{-1}</math> --------- We know that <math>A^{-1}I=A^{-1}</math> | |||
<math>IB(AB)^{-1}=A^{-1}</math> ---------We know that <math>A^{-1}A=I</math> | |||
<math>B(AB)^{-1}=A^{-1}</math>---------We know that <math>IB=B</math> | |||
<math>B^{-1}B(AB)^{-1}=B^{-1}A^{-1}</math> --------- Multiply by <math>B^{-1}</math> on both sides | |||
<math>I(AB)^{-1}=B^{-1}A^{-1}</math>---------We know that <math>B^{-1}B=I</math> | |||
<math>(AB)^{-1}=B^{-1}A^{-1}</math>---------We know that <math>I(AB)^{-1}=(AB)^{-1}</math> | |||
== Applications of Invertible Matrix == | |||
* Invertible matrices can be used to encrypt a message. There are many ways to encrypt a message and the use of coding has become particularly significant in recent years. | |||
* Invertible matrices are employed by cryptographers to decode a message as well, especially those programming the specific encryption algorithm. | |||
* Computer graphics in the 3D space use invertible matrices to render what you see on the screen. |
Latest revision as of 07:20, 12 January 2024
In linear algebra, an square matrix is called invertible, if the product of the matrix and its inverse is the identity matrix.
Definition
A matrix of dimension is called invertible if and only if there exists another matrix of the same dimension, such that , where is the identity matrix of the same order. Matrix is known as the inverse of matrix . Inverse of matrix is symbolically represented by.Invertible matrix is also known as a non-singular matrix or non-degenerate matrix.
For example, matrices and are given below:
Now we multiply with and obtain an identity matrix:
Similarly, on multiplying B with A, we obtain the same identity matrix:
We can that
Hence and is known as the inverse of
and can also be called an inverse of
Invertible Matrices Theorem
Theorem 1
If there exists an inverse of a square matrix, it is always unique.
Proof:
Let is a square matrix of order . Let matrices and to be inverses of matrix .
Now since is the inverse of matrix .
Similarly, .
But
This proves or and are the same matrices.
Theorem 2
If and are matrices of the same order and are invertible, then
Proof:
As per the definition of inverse of a matrix
--------- Multiply by on both sides
--------- We know that
---------We know that
---------We know that
--------- Multiply by on both sides
---------We know that
---------We know that
Applications of Invertible Matrix
- Invertible matrices can be used to encrypt a message. There are many ways to encrypt a message and the use of coding has become particularly significant in recent years.
- Invertible matrices are employed by cryptographers to decode a message as well, especially those programming the specific encryption algorithm.
- Computer graphics in the 3D space use invertible matrices to render what you see on the screen.