## ----nomessages, echo = FALSE-------------------------------------------------
knitr::opts_chunk$set(
  warning = FALSE,
  message = FALSE
)
options(digits=4)
par(mar=c(5,4,1,1)+.1)

## -----------------------------------------------------------------------------
	library(matlib)

## -----------------------------------------------------------------------------
  A <- matrix(c(4, 2, 1,
                5, 6, 7,
                1, 0, 3), nrow=3, byrow=TRUE)
  det(A)

## -----------------------------------------------------------------------------
  cat(cofactor(A, 1, 1),  "  ==  ",  1 * det( (A[-1, -1]), "\n" ))
  cat(cofactor(A, 1, 2),  "  ==  ", -1 * det( (A[-1, -2]), "\n" ))
  cat(cofactor(A, 1, 3),  "  ==  ",  1 * det( (A[-1, -3]), "\n" ))

## -----------------------------------------------------------------------------
  rowCofactors(A, 1)

## -----------------------------------------------------------------------------
  A[1,] %*% rowCofactors(A, 1)
  all.equal( det(A), c(A[1,] %*% rowCofactors(A, 1)) )

## -----------------------------------------------------------------------------
  M <- matrix(c(2, 3, 1, 2,
                4, 2, 3, 4,
                1, 4, 2, 2,
                3, 1, 0, 1), nrow=4, ncol=4, byrow=TRUE)
  (dsave <- det(M))

# ### 'pivot' on the leading diagonal element, M[1,1]:

## -----------------------------------------------------------------------------
  (d <- M[1,1])

    #-- Reduce row 1, col 1 to 0
  (M[1,] <- M[1,, drop=FALSE] / M[1, 1])
  (M <- M - M[,1] %*%  M[1,, drop=FALSE])

    #-- Drop first row and column
  M <- M[-1, -1]

    #-- Accumulate the product of pivots
  d <- d * M[1, 1]

## -----------------------------------------------------------------------------
  (M[1,] <- M[1,, drop=FALSE] / M[1,1])
  (M <- M - M[,1] %*%  M[1,, drop=FALSE])
  M <- M[-1, -1]
  d = d * M[1, 1]

## -----------------------------------------------------------------------------
  (M[1,] <- M[1,, drop=FALSE] / M[1,1])
  (M <- M - M[,1] %*%  M[1,, drop=FALSE])
  M <- M[-1, -1, drop=FALSE]
  d <- d * M[1, 1]

  # did we get it right?
  all.equal(d, dsave)