Definitions of Inverse Matrix and Proof of Inverse Matrix Formulas

In this article, we will discuss the definition of inverse matrix that solve using elementary row operations and apply these operations simultaneously to In to get A-1.

Often it will not be known in advance whether a matrix can be reversed. If the matrix cannot be reversed, then the matrix is ​​in the form of a reduced line echelon that has at least a zero number row and will appear on the left hand side. So, if the given matrix cannot be reversed, the calculation can be stopped.

In connection with that, the method that will be used to perform the procedure is the Gauss-Jordan elimination method. This method will increase the superiority of its application even though the given matrix cannot be reversed.


Definition of inverse matrix

A quadratic matrix A has the n expressed as



It is called having an inverse if there is a matrix B, so AB=BA=In, then A is said to be invertible. B matrix is ​​called inverse matrix A (invertible), written A-1, is a quadratic matrix with n.

Example 1.
Calculate the inverse of 


Solution:
For Example,  

When multiplied it will be obtained:


So that the equation produced is


To get the values ​​of a1, a2, a3, and a4 then the suitable method is the Gauss-Jordan elimination method which will add the first line to the second row,

First column,
Get a value of a1,

Get a3 value by substituting the value of a1 previously obtained in equation 1 or 2.

Second Column,
Get a value of a2,

Get a4 value by substituting the value of a2 previously obtained in equation 3 or 4.

From the above calculations obtained a1 = 3/2, a2 ​​= -1/2, a3 = -2, and a4 = 1.

Example 2.
Find the inverse value of matrix A with 2 x 2 below,


Then the calculation is as follows, It is known that,


Finding the value of Adj(A) from matrix A with the first step is to find the value of Cofactor Cij.


From the calculation of the cofactor above, the value of Adj(A) is


then,


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