Matrix Inverse
If A is a non-singular square matrix, there is an existence of n x n matrix A^{-1}, which is called the inverse matrix of A such that it satisfies the property:
AA^{-1} = A^{-1}A = I, where I is Â the Identity matrix
The identity matrix for the 2 x 2 matrix is given by
It is noted that in order to find the inverse matrix, the square matrix should be non-singular whose determinant value does not equal to zero.
Let us take the square matrix A
Where a, b, c, and d represents the number.
The determinant of the matrix A is written as ad-bc, where the value is not equal to zero. The inverse matrix can be found for 2Ã—Â 2, 3Ã— 3, …n Ã— n matrices. Finding the inverse of a 3Ã—3 matrix is a bit difficult than finding the inverses of a 2 Ã—2 matrices.
Inverse Matrix Method
The inverse of a matrix Â can be found using the following methods:
Method 1:
Similarly, we can find the inverse of a 3Ã—3 matrix by finding the determinant value of the given matrix.
Check out: Inverse matrix calculator
Method 2:
- The inverse matrix is also found using the following equation:
A^{-1}= adj(A)/det(A),
Â Â Â Â Â whereÂ adj(A) refers to the adjoint of a matrix A,Â det(A) refers to the determinant of a matrix A.
- The adjoint of a matrix A or adj(A) can be found using the following method.
Â Â Â Â Â In order to find the adjoint of a matrix A first, find the cofactor matrix of a given matrix and thenÂ Â
Â Â Â Â Â take the transpose of a cofactor matrix.
- The cofactor of a matrix can be obtained as
C_{ij}Â = (-1)^{ij}Â det (Mij)
Here, Mij refers to the (i,j)^{th}Â minor matrix after removing the ith row and the jth column. You can also say that the transpose of a cofactor matrix is also called as the adjoint of a matrix A.
Similarly, we can also find the inverse of a 3 x 3 matrix. Here also the first step would be to find the determinant, followed by the next step – Transpose.
Method 3:
Finding an Inverse Matrix by Elementary Transformation
Let us consider three matrices X, A and B such that X = AB. To determine the inverse of a matrix using elementary transformation, we convert the given matrix into an identity matrix.
If the inverse of matrix A, A^{-1} exists then to determine A^{-1} using elementary row operations
- Write A = IA, where I is the identity matrix of the same order as A.
- Apply a sequence of row operations till we get identity matrix on the LHS and use the same elementary operations on the RHS to get I = BA. The matrix B on the RHS is the inverse of matrix A.
- To find the inverse of A using column operations, write A = IA and apply column operations sequentially till I = AB is obtained, where B is the inverse matrix of A.
Inverse Matrix Example
To understand this concept better let us take a look at the following example.
Example: Find the inverse of matrix A given below:
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