Difference between revisions of "Invertible Matrices"
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== Definition == | == Definition == | ||
A matrix <math>A</math> of dimension <math>n \times n</math> is called invertible if and only if there exists another matrix <math>B</math> of the same dimension, such that <math>AB=BA=I</math>, where <math>I</math> is the identity matrix of the same order. Matrix <math>B</math> is known as the inverse of matrix <math>A</math>. Inverse of matrix <math>A</math> is symbolically represented by<math>A^{-1}</math>.Invertible matrix is also known as a non-singular matrix or non-degenerate matrix. | A matrix <math>A</math> of dimension <math>n \times n</math> is called invertible if and only if there exists another matrix <math>B</math> of the same dimension, such that <math>AB=BA=I</math>, where <math>I</math> is the identity matrix of the same order. Matrix <math>B</math> is known as the inverse of matrix <math>A</math>. Inverse of matrix <math>A</math> is symbolically represented by<math>A^{-1}</math>.Invertible matrix is also known as a non-singular matrix or non-degenerate matrix. | ||
For example, matrices <math>A</math> and <math>B</math> are given below: | |||
<math>A=\begin{bmatrix} 1 & 2 \\ 2 & 5 \end{bmatrix}</math> <math>B=\begin{bmatrix} 5 & -2 \\ -2 & 1 \end{bmatrix}</math><math>A=\begin{bmatrix} 1 & 2 \\ 2 & 5 \end{bmatrix}</math> | |||
Now we multiply <math>A</math> with <math>B</math> and obtain an identity matrix: | |||
<math>AB=\begin{bmatrix} 1 \times 5 + 2 \times -2 & 1 \times -2+2 \times 1 \\ 2 \times 5 + 5 \times -2 & 2 \times -2+5 \times 1 \end{bmatrix}=\begin{bmatrix} 5-4 & -2+2 \\ 10 -10 & -4+5 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}</math> | |||
Similarly, on multiplying B with A, we obtain the same identity matrix: | |||
<math>BA=\begin{bmatrix} 5 \times 1+ -2 \times 2 & 5\times 2+ -2 \times 5 \\ -2 \times 1 + 1 \times 2 & 2 \times -2+1 \times 5 \end{bmatrix}=\begin{bmatrix} 5-4 & 10-10 \\ -2+2 & -4+5 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}</math> | |||
We can that <math>AB=BA=I</math> | |||
Hence <math>A^{-1}=B</math> and <math>B</math> is known as the inverse of <math>A</math> | |||
<math>B^{-1}=A</math> and <math>A</math> can also be called an inverse of <math>B</math> |
Revision as of 09:42, 8 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