In exams: Automatic Generation of Exams in R

## DATA GENERATION
## number of rows/columns
n <- sample(3:4, 1)
## elements on lower triangle (and diagonal)
m <- n * (n + 1)/2
L <- matrix(data = 0, nrow = n, ncol = n)
diag(L) <- sample(1:5, n, replace = TRUE)
L[lower.tri(L)] <- sample(-5:5, m-n, replace = TRUE)
## matrix A for which the Cholesky decomposition should be computed
A <- L %*% t(L)

## rnadomly generate questions/solutions/explanations
mc <- matrix_to_mchoice(
  L,                                     ## correct matrix
  y = sample(-10:10, 5, replace = TRUE), ## random values for comparison
  lower = TRUE,                          ## only lower triangle/diagonal
  name = "\\ell",                        ## name for matrix elements
  restricted = TRUE)                     ## assure at least one correct and one wrong solution

Question

For the matrix $$ \begin{aligned} A &= r toLatex(A, escape = FALSE). \end{aligned} $$ compute the matrix $L = (\ell_{ij})_{1 \leq i,j \leq r n}$ from the Cholesky decomposition $A = L L^\top$.

Which of the following statements are true?

answerlist(mc$questions, markup = "markdown")

Solution

The decomposition yields $$ \begin{aligned} L &= r toLatex(L, escape = FALSE) \end{aligned} $$ and hence:

answerlist(
  ifelse(mc$solutions, "True", "False"),
  mc$explanations, markup = "markdown")

Meta-information

extype: mchoice exsolution: r mchoice2string(mc$solutions) exname: Cholesky decomposition

exams documentation built on Nov. 14, 2022, 3:02 p.m.