This another method to compute the inverse of matrix. The method consist of the

Question

This another method to compute the inverse of matrix. The method consist of the following steps: Step 1: State (A|Id)). Step 2: The idea now is to convert A into Id by performing valid transformations (the same you use to reduce a system to its skelon form). Thus, each transformation you perform on A (left side) in order to convert it into the identity matrix Id, you must perform it on Id (right side). Not only A is going to change, Id will be changed as well. Step 3: When you get A converted into Id in the left side, Id will be converted into A^-1 in the right side. (a) Prove that each valid transformation by multiplying by some matrix. Specify the matrix in each case. (b) Why does this method work? Explain. (c) Apply this method to matrices given in the previous exercise.

Explanation / Answer

Let A be an nxn non singular matrix

then A = I A where I is the unit matrix

Applying series of elemeentary row operations on A to reduce I and applying the same row operation in I in RHS we get ' I = B.A

ie the matrix B is obtained from I by applying the same elementary operations and B is the inverse of A

ie to find the inverse of the non singular matrix A

we apply series of elementary operations to reduce A to I

the same operations applied to I will give inverse of A

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