Author: Michael Friendly
Date: 2025-10-05

# 1. Properties of determinants

#### Michael Friendly

#### 2025-10-05

The following examples illustrate the basic properties of the determinant of a matrix. We do this first with simple numerical examples and then using geometric diagrams.

### Create a 2 x 2 matrix

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

    ##      [,1] [,2]
    ## [1,]    3    1
    ## [2,]    2    4

    det(A)

    ## [1] 10

### 1. Interchange two rows or cols changes the sign: -> -1 * det(A)

    det(A[ 2:1, ])

    ## [1] -10

    det(A[, 2:1 ])

    ## [1] -10

### 2. transpose -> det (A) unchanged

    det( t(A) )

    ## [1] 10

### 3. multiply row * k -> k * det(A)

Note that to multiply rows by different constants requires a diagonal matrix on the left.

    diag(c(3, 1)) %*% A

    ##      [,1] [,2]
    ## [1,]    9    3
    ## [2,]    2    4

    det( diag(c(3, 1))  %*% A)

    ## [1] 30

### 4. multiply matrix * k -> k^2 * det(A)

This is because multiplying a matrix by a constant multiplies each row.

    det(3 * A)

    ## [1] 90

    3^2 * det(A)

    ## [1] 90

### 5. det (A B) -> det(A) * det(B)

The determinant of a product is the product of the determinants. The same holds for any number of terms in a matrix product.

    B <- matrix(c(4, 2,
                 3, 5), nrow=2, byrow=TRUE)
    B

    ##      [,1] [,2]
    ## [1,]    4    2
    ## [2,]    3    5

    det(A %*% B)

    ## [1] 140

    det(A) * det(B)

    ## [1] 140

### 6. proportional rows or columns -> `det() == 0`

Here we just add an additional copy of column 1 of a matrix, so `C[,3] == C[,1]`. The determinant is 0 because the columns are linearly dependent.

    C <- matrix(c(1, 5,
                 2, 6,
                 4, 4), nrow=3, byrow=TRUE)
    C <- cbind(C, C[,1])
    C

    ##      [,1] [,2] [,3]
    ## [1,]    1    5    1
    ## [2,]    2    6    2
    ## [3,]    4    4    4

    det(C)

    ## [1] 0

### 7. Add multiple of one row to another -> det unchanged

This is the principle behind one of the elementary row operations.

    A[2,] <- A[2,] - 2*A[1,]
    det(A)

    ## [1] 10

### 8. Geometric interpretation

Many aspects of matrices and vectors have geometric interpretations. For \(2 \times 2\) matrices, the determinant is the area of the parallelogram defined by the rows (or columns), plotted in a 2D space. (For \(3 \times 3\) matrices, the determinant is the volume of a parallelpiped in 3D space.)

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

    ##      [,1] [,2]
    ## [1,]    3    1
    ## [2,]    2    4

    det(A)

    ## [1] 10

The `matlib` package has some handy functions (`vectors()`) for drawing geometric diagrams.

    library(matlib)
    xlim <- c(0,6)
    ylim <- c(0,6)
    par(mar=c(3,3,1,1)+.1)
    plot(xlim, ylim, type="n", xlab="X1", ylab="X2", asp=1)
    sum <- A[1,] + A[2,]
    # draw the parallelogram determined by the rows of A
    polygon( rbind(c(0,0), A[1,], sum, A[2,]), col=rgb(1,0,0,.2))
    vectors(A, labels=c("a1", "a2"), pos.lab=c(4,2))
    vectors(sum, origin=A[1,], col="gray")
    vectors(sum, origin=A[2,], col="gray")
    # add some annotations
    text(0,6, "det(A) is the area of its row vectors", pos=4)
    text(mean(A[,1]), mean(A[,2]), "det(A)", cex=1.25)

There is a simple visual proof of this fact about determinants but it is easiest to see in the case of a diagonal matrix, where the row vectors are orthogonal, so area is just height x width.

    (D <- 2 * diag(2))

    ##      [,1] [,2]
    ## [1,]    2    0
    ## [2,]    0    2

    det(D)

    ## [1] 4

Plot this as before:

    par(mar=c(3,3,1,1)+.1)
    plot(c(0,2), c(0,2), type="n", xlab="X1", ylab="X2", asp=1)
    sum <- D[1,] + D[2,]
    polygon( rbind(c(0,0), D[1,], sum, D[2,]), col=rgb(0,1,0,.2))
    vectors(D, labels=c("d1", "d2"), pos.lab=c(3,4))
    vectors(sum, origin=D[1,], col="gray")
    vectors(sum, origin=D[2,], col="gray")
    text(mean(D[,1]), mean(D[,2]), "det(D)", cex=1.25)

Finally, we can also see why the determinant is zero when the rows or columns are proportional.

    (B <- matrix(c(1, 2, 2, 4), 2,2))

    ##      [,1] [,2]
    ## [1,]    1    2
    ## [2,]    2    4

    det(B)

    ## [1] 0

Such vectors are called collinear. They enclose no area.

    par(mar=c(3,3,1,1)+.1)
    plot(c(0,4), c(0,4), type="n", xlab="X1", ylab="X2", asp=1)
    vectors(B, labels=c("b1", "b2"), pos.lab=c(4,2))