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Inverse of a Matrix - Inverse of a Nonsingular Matrix by Elementary Transformation

notes

Let X, A and B be matrices of, the same order such that X = AB. In order to apply a sequence of elementary row operations on the matrix equation X = AB, we will apply these row operations simultaneously on X and on the first matrix A of the product AB on RHS. 
Similarly, in order to apply a sequence of elementary column operations on the matrix equation X = AB, we will apply, these operations simultaneously on X and on the second matrix B of the product AB on RHS.
In view of the above discussion, we conclude that if A is a matrix such that `A^(–1)` exists, then to find `A^(–1)` using elementary row operations, write A = IA and apply a sequence of row operation on A = IA till we get, I = BA. The matrix B will be the inverse of A. Similarly, if we wish to find `A^(–1)` using column operations, then, write A = AI and apply a sequence of column operations on A = AI till we get, I = AB.
Remark: In case, after applying one or more elementary row (column) operations on A = IA (A = AI), if we obtain all zeros in one or more rows of the matrix A on L.H.S., then `A^(–1)` does not exist.

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Principle of mathematical induction [00:35:45]
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