{r data generation, echo = FALSE, results = "hide"} ## 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? {r questionlist, echo = FALSE, results = "asis"} answerlist(mcquestions, markup = "markdown")  Solution ======== The decomposition yields \begin{aligned} L &= r toLatex(L, escape = FALSE) \end{aligned} and hence: {r solutionlist, echo = FALSE, results = "asis"} answerlist( ifelse(mcsolutions, "True", "False"), mc$explanations, markup = "markdown")  Meta-information ================ extype: mchoice exsolution: r mchoice2string(mc$solutions) exname: Cholesky decomposition