Terms Related to Matrix Inverse
The terminology listed below can help you grasp the inverse of a matrix more clearly and easily.
| Terms | Definition | Formula/Process | Example with Matrix A |
|---|---|---|---|
| Minor | The minor of an element in a matrix is the determinant of the matrix formed by removing the row and column of that element. | For element aij , remove the ith row and jth column to form a new matrix and find its determinant. | Minor of a11 is the determinant of [Tex]A = \begin{bmatrix}5 & 6\\ 6 & 7\end{bmatrix} [/Tex] |
| Cofactor | The cofactor of an element is the minor of that element multiplied by (-1)i+j , where i and j are the row and column indices of the element. | Cofactor of aij = (-1)i+j Minor of aij | Cofactor of a11 = (-1)1+1 × Minor of a11 = Minor of a11 |
| Determinant | The determinant of a matrix is calculated as the sum of the products of the elements of any row or column and their respective cofactors. | For a row (or column), sum up the product of each element and its cofactor. | Determinant of A = a11 × Cofactor of a11 +a12 × Cofactor of a12 +a13 × Cofactor of a13. |
| Adjoint | The adjoint of a matrix is the transpose of its cofactor matrix. | Create a matrix of cofactors for each element of the original matrix and then transpose it. | Adjoint of A is the transpose of the matrix formed by the cofactors of all elements in A. |
Singular Matrix
A matrix whose value of the determinant is zero is called a singular matrix, i.e. any matrix A is called a singular matrix if |A| = 0. Inverse of a singular matrix does not exist.
Non-Singular Matrix
A matrix whose value of the determinant is non-zero is called a non-singular matrix, i.e. any matrix A is called a non-singular matrix if |A| ≠ 0. Inverse of a non-singular matrix exists.
Identity Matrix
A square matrix in which all the elements are zero except for the principal diagonal elements is called the identity matrix. It is represented using I. It is the identity element of the matrix as for any matrix A,
A×I = A
An example of an Identity matrix is,
I3×3 =[Tex] \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{bmatrix}[/Tex]
This is an identity matrix of order 3×3.\
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Inverse of a Matrix
The inverse of Matrix is the matrix that on multiplying with the original matrix results in an identity matrix. For any matrix A, its inverse is denoted as A-1.
Let’s learn about the Matrix Inverse in detail, including its definition, formula, methods on how to find the inverse of a matrix, and examples.
Table of Content
- Matrix Inverse
- Terms Related to Matrix Inverse
- How to Find Inverse of Matrix?
- Inverse of a Matrix Formula
- Inverse Matrix Method
- Inverse of 2×2 Matrix Example
- Determinant of Inverse Matrix
- Properties of Inverse of Matrix
- Matrix Inverse Solved Examples