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.
rowadd(x, from, to, mult)a numeric matrix, possibly consisting of the coefficient matrix, A, joined with a vector of constants, b.
the index of one or more source rows. If from is a vector, it must have the same length as to.
the index of one or more destination rows
the multiplier(s)
the matrix x, as modified
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.
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
#> b
#> [1,] 2 1.0 -1.0 8
#> [2,] 0 0.5 0.5 1
#> [3,] -2 1.0 2.0 -3
(Ab <- rowadd(Ab, 1, 3, 1)) # row 3 <- row 3 + 1 row 1
#> b
#> [1,] 2 1.0 -1.0 8
#> [2,] 0 0.5 0.5 1
#> [3,] 0 2.0 1.0 5
(Ab <- rowadd(Ab, 2, 3, -4)) # row 3 <- row 3 - 4 row 2
#> b
#> [1,] 2 1.0 -1.0 8
#> [2,] 0 0.5 0.5 1
#> [3,] 0 0.0 -1.0 1
# multiply to make diagonals = 1
(Ab <- rowmult(Ab, 1:3, c(1/2, 2, -1)))
#> b
#> [1,] 1 0.5 -0.5 4
#> [2,] 0 1.0 1.0 2
#> [3,] 0 0.0 1.0 -1
# The matrix is now in triangular form
# Could continue to reduce above diagonal to zero
echelon(A, b, verbose=TRUE, fractions=TRUE)
#>
#> Initial matrix:
#> [,1] [,2] [,3] [,4]
#> [1,] 2 1 -1 8
#> [2,] -3 -1 2 -11
#> [3,] -2 1 2 -3
#>
#> row: 1
#>
#> exchange rows 1 and 2
#> [,1] [,2] [,3] [,4]
#> [1,] -3 -1 2 -11
#> [2,] 2 1 -1 8
#> [3,] -2 1 2 -3
#>
#> multiply row 1 by -1/3
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 2 1 -1 8
#> [3,] -2 1 2 -3
#>
#> multiply row 1 by 2 and subtract from row 2
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 0 1/3 1/3 2/3
#> [3,] -2 1 2 -3
#>
#> multiply row 1 by 2 and add to row 3
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 0 1/3 1/3 2/3
#> [3,] 0 5/3 2/3 13/3
#>
#> row: 2
#>
#> exchange rows 2 and 3
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 0 5/3 2/3 13/3
#> [3,] 0 1/3 1/3 2/3
#>
#> multiply row 2 by 3/5
#> [,1] [,2] [,3] [,4]
#> [1,] 1 1/3 -2/3 11/3
#> [2,] 0 1 2/5 13/5
#> [3,] 0 1/3 1/3 2/3
#>
#> multiply row 2 by 1/3 and subtract from row 1
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 -4/5 14/5
#> [2,] 0 1 2/5 13/5
#> [3,] 0 1/3 1/3 2/3
#>
#> multiply row 2 by 1/3 and subtract from row 3
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 -4/5 14/5
#> [2,] 0 1 2/5 13/5
#> [3,] 0 0 1/5 -1/5
#>
#> row: 3
#>
#> multiply row 3 by 5
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 -4/5 14/5
#> [2,] 0 1 2/5 13/5
#> [3,] 0 0 1 -1
#>
#> multiply row 3 by 4/5 and add to row 1
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 0 2
#> [2,] 0 1 2/5 13/5
#> [3,] 0 0 1 -1
#>
#> multiply row 3 by 2/5 and subtract from row 2
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 0 2
#> [2,] 0 1 0 3
#> [3,] 0 0 1 -1