Mathematics ELEMENTARY TRANSFORMATIONS

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ELEMENTARY TRANSFORMATIONS

Elementary Operation (Transformation) of a Matrix

There are six operations (transformations) on a matrix, three of which are due to rows and three due to columns, which are known as elementary operations or transformations.

(i) The interchange of any two rows or two columns. Symbolically the interchange of `i^(th)` and `j^(th)` rows is denoted by `R_i ↔ R_j` and interchange of `i^(th)` and `j^(th)` column is denoted by `C_i ↔ C_j`.

For example, applying `R_1↔ R_2` to `A = [(1,2,1),(-1,sqrt3,1),(5,6,7)]`, we get `[(-1,sqrt3,1),(1,2,1),(5,6,7)]`

(ii) The multiplication of the elements of any row or column by a non zero number. Symbolically, the multiplication of each element of the ith row by `k`,

where ` k ≠ 0` is denoted by `R_i → k R_i`.

The corresponding column operation is denoted by `C_i → kC_i`

For example, applying `C_3 -> 1/7 C_3`, to `B = [(1,2,1),(-1,sqrt3,1)]`, we get `[(1,2,1/7),(-1,sqrt3, 1/7)]`

(iii) The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non zero number.

Symbolically, the addition to the elements of ith row, the corresponding elements of `j^(th)` row multiplied by `k` is denoted by `R_i --> R_i + kR_j`.

The corresponding column operation is denoted by `C_i → C_i + kC_j`.

For example, applying `R_1 -> R_2-2R_1` to `C = [(1,2),(2,-1)]`, we get `[(1,2),(0,-5)]`

There are six operations (transformations) on a matrix, three of which are due to rows and three due to columns, which are known as elementary operations or transformations.

(i) The interchange of any two rows or two columns. Symbolically the interchange of `i^(th)` and `j^(th)` rows is denoted by `R_i ↔ R_j` and interchange of `i^(th)` and `j^(th)` column is denoted by `C_i ↔ C_j`.

For example, applying `R_1↔ R_2` to `A = [(1,2,1),(-1,sqrt3,1),(5,6,7)]`, we get `[(-1,sqrt3,1),(1,2,1),(5,6,7)]`

(ii) The multiplication of the elements of any row or column by a non zero number. Symbolically, the multiplication of each element of the ith row by `k`,

where ` k ≠ 0` is denoted by `R_i → k R_i`.

The corresponding column operation is denoted by `C_i → kC_i`

For example, applying `C_3 -> 1/7 C_3`, to `B = [(1,2,1),(-1,sqrt3,1)]`, we get `[(1,2,1/7),(-1,sqrt3, 1/7)]`

(iii) The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non zero number.

Symbolically, the addition to the elements of ith row, the corresponding elements of `j^(th)` row multiplied by `k` is denoted by `R_i --> R_i + kR_j`.

The corresponding column operation is denoted by `C_i → C_i + kC_j`.

For example, applying `R_1 -> R_2-2R_1` to `C = [(1,2),(2,-1)]`, we get `[(1,2),(0,-5)]`

Inverse of a matrix by elementary operations

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.

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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