rowadd: Elementary Row Operations

Description

The elementary row operation rowadd adds multiples of one or more rows to other rows of a matrix. This is usually used as a means to solve systems of linear equations, of the form \(A x = b\), and rowadd corresponds to adding equals to equals.

Usage

rowadd(x, from, to, mult)

Value

the matrix x, as modified

Arguments

x: a numeric matrix, possibly consisting of the coefficient matrix, A, joined with a vector of constants, b.
from: the index of one or more source rows. If from is a vector, it must have the same length as to.
to: the index of one or more destination rows
mult: the multiplier(s)

Details

The functions rowmult and rowswap complete the basic operations used in reduction to row echelon form and Gaussian elimination. These functions are used for demonstration purposes.

Examples

Run this code

A <- matrix(c(2, 1, -1,
             -3, -1, 2,
             -2,  1, 2), 3, 3, byrow=TRUE)
b <- c(8, -11, -3)

# using row operations to reduce below diagonal to 0
Ab <- cbind(A, b)
(Ab <- rowadd(Ab, 1, 2, 3/2))  # row 2 <- row 2 + 3/2 row 1
(Ab <- rowadd(Ab, 1, 3, 1))    # row 3 <- row 3 + 1 row 1
(Ab <- rowadd(Ab, 2, 3, -4))   # row 3 <- row 3 - 4 row 2
# multiply to make diagonals = 1
(Ab <- rowmult(Ab, 1:3, c(1/2, 2, -1)))
# The matrix is now in triangular form

# Could continue to reduce above diagonal to zero
echelon(A, b, verbose=TRUE, fractions=TRUE)

# convenient use of pipes
I <- diag( 3 )
AA <- I |>
  rowadd(3, 1, 1) |>   # add 1 x row 3 to row 1
  rowadd(1, 3, 1) |>   # add 1 x row 1 to row 3
  rowmult(2, 2)        # multiply row 2 by 2