# Matrix inversion by elementary row operations" In matlib: Matrix Functions for Teaching and Learning Linear Algebra and Multivariate Statistics

```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):

• Add a multiple of one row to another (`rowadd()`)
• Multiply one row by a constant (`rowmult()`)
• Interchange two rows (`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:

1. Adjoin the identity matrix to the right side of A, to give the matrix \$[A | I]\$
2. Apply row operations to this matrix until the left (\$A\$) side is reduced to \$I\$
3. The inverse matrix appears in the right (\$I\$) side

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.

### Load the `matlib` package

```library(matlib)
```

### Create a 3 x 3 matrix

```   A <- matrix( c(1, 2, 3,
2, 3, 0,
0, 1,-2), nrow=3, byrow=TRUE)
```

### Join an identity matrix to A

```   (AI <-  cbind(A, diag(3)))
```

### Apply elementary row operations to reduce A to an identity matrix.

The right three cols will then contain inv(A). We will do this three ways:

1. first, just using R arithmetic on the rows of `AI`
2. using the ERO functions in the `matlib` package
3. using the `echelon()` function

### 1. Using R arithmetic

```    (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)
```

### 2. Do the same, using matlib functions `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
```

### 3. Using `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)
```

## Try the matlib package in your browser

Any scripts or data that you put into this service are public.

matlib documentation built on April 4, 2018, 5:03 p.m.