knitr::opts_chunk$set( warning = FALSE, message = FALSE ) options(digits=4) par(mar=c(5,4,1,1)+.1)
The following examples illustrate the steps in finding the inverse of a matrix using elementary row operations (EROs):
rowadd()
)rowmult()
)rowswap()
)These have the properties that they do not change the inverse. The method used here is sometimes called the Gauss-Jordan method, a form of Gaussian elimination. Another term is (row-reduced) echelon form.
Steps:
Why this works: The series of row operations transforms $$ [A | I] \Rightarrow [A^{-1} A | A^{-1} I] = [I | A^{-1}]$$
If the matrix is does not have an inverse (is singular) a row of all zeros will appear in the left ($A$) side.
matlib
packagelibrary(matlib)
A <- matrix( c(1, 2, 3, 2, 3, 0, 0, 1,-2), nrow=3, byrow=TRUE)
(AI <- cbind(A, diag(3)))
The right three cols will then contain inv(A). We will do this three ways:
AI
matlib
packageechelon()
function(AI[2,] <- AI[2,] - 2*AI[1,]) # row 2 <- row 2 - 2 * row 1 (AI[3,] <- AI[3,] + AI[2,]) # row 3 <- row 3 + row 2 (AI[2,] <- -1 * AI[2,]) # row 2 <- -1 * row 2 (AI[3,] <- -(1/8) * AI[3,]) # row 3 <- -.25 * row 3
Now, all elements below the diagonal are zero
AI #--continue, making above diagonal == 0 AI[2,] <- AI[2,] - 6 * AI[3,] # row 2 <- row 2 - 6 * row 3 AI[1,] <- AI[1,] - 3 * AI[3,] # row 1 <- row 1 - 3 * row 3 AI[1,] <- AI[1,] - 2 * AI[2,] # row 1 <- row 1 - 2 * row 2 AI #-- last three cols are the inverse (AInv <- AI[,-(1:3)]) #-- compare with inv() inv(A)
rowadd()
, rowmult()
and rowswap()
AI <- cbind(A, diag(3)) AI <- rowadd(AI, 1, 2, -2) # row 2 <- row 2 - 2 * row 1 AI <- rowadd(AI, 2, 3, 1) # row 3 <- row 3 + row 2 AI <- rowmult(AI, 2, -1) # row 1 <- -1 * row 2 AI <- rowmult(AI, 3, -1/8) # row 3 <- -.25 * row 3 # show result so far AI #--continue, making above-diagonal == 0 AI <- rowadd(AI, 3, 2, -6) # row 2 <- row 2 - 6 * row 3 AI <- rowadd(AI, 2, 1, -2) # row 1 <- row 1 - 2 * row 2 AI <- rowadd(AI, 3, 1, -3) # row 1 <- row 1 - 3 * row 3 AI
echelon()
echelon()
does all these steps row by row, and returns the result
echelon( cbind(A, diag(3)))
It is more interesting to see the steps, using the argument verbose=TRUE
. In
many cases, it is informative to see the numbers printed as fractions.
echelon( cbind(A, diag(3)), verbose=TRUE, fractions=TRUE)
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