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Question

How to find the inverse of a matrix?


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Solution

Step1. Inverse of a matrix

Let a square matrix of order n×n, denoted by A then it's inverse is denoted by A-1but it not written as A-1≠1A

Then A-1=1A.adj(A), where C is determinant of given matrix.

Step2. Find the matrix of minor.

Consider, matrix A=a11a12a13a21a22a23a31a32a33

Then, Minors of A is given by

M11=a22a23a32a33

For calculating minors of the matrix just the hide the row and column for which minors to be calculated.

Step3. Change of matrix into cofactors.
Cofactor of matrix is

Cij=(-1)i+jMi+j

C11=(-1)2M11=1Ă—(a22Ă—a33-a23Ă—a32)=(a22Ă—a33-a23Ă—a32)

Step4. Multiply by the reciprocal of determinant

Determinant of matrix is Aand calculated as

A=a11a12a13a21a22a23a31a32a33=a11(a22Ă—a33-a23Ă—a32)-a12(a21Ă—a33-a23a31)+a13(a21Ă—a32-a22Ă—a31)

Step5. Ad joint of a matrix

Ad joint of matrix is transpose of cofactor of matrix.

Transpose of cofactor =C11C12C13C21C22C23C31C32C33

Then,

CT=C11C21C31C12C22C32C13C23C33

Therefore,

A-1=AdjAA=1AC11C21C31C12C22C32C13C23C33

Hence, in this way we can find the inverse of a matrix.


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