# Properties of determinants" In matlib: Matrix Functions for Teaching and Learning Linear Algebra and Multivariate Statistics

knitr::opts_chunk$set( warning = FALSE, message = FALSE, fig.height = 5, fig.width = 5 ) options(digits=4) par(mar=c(3,3,1,1)+.1)  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 det(A)  ### 1. Interchange two rows or cols changes the sign: -> -1 * det(A) det(A[ 2:1, ]) det(A[, 2:1 ])  ### 2. transpose -> det (A) unchanged det( t(A) )  ### 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 det( diag(c(3, 1)) %*% A)  ### 4. multiply matrix * k -> k^2 * det(A) This is because multiplying a matrix by a constant multiplies each row. det(3 * A) 3^2 * det(A)  ### 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 det(A %*% B) det(A) * det(B)  ### 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 det(C)  ### 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)  ### 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 parallelepiped in 3D space.)

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


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")
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))
det(D)


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))
det(B)


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))


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