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3 Functions for Matrices

The determinant of a square matrix can be computed in R by using the function det for instance

: det(E)

The transpose of a matrix can be obtained by using t() e.g.

: t(E)

The inverse of a square matrix can be obtained by using the function solve

: solve(E)

Quiz 1: Inverting a $5\times 5$ matrix

Find the inverse of the matrix

\displaystyle\left[\begin{array}[]{ccccc}1&-1&0&3&1\\ 0&-1&0&1&2\\ 1&1&0&1&-1\\ 0&2&1&1&1\\ -1&0&0&0&2\\ \end{array}\right]

(A)

$\displaystyle\left[\begin{array}[]{ccccc}-1&2&1&0&-1\\ -1/4&0&3/4&0&1/2\\ 1/4&0&-7/4&1&-3/2\\ 3/4&1&1/4&0&1/2\\ -1/2&1&-1/2&0&0\\ \end{array}\right]$
(B)

$\displaystyle\left[\begin{array}[]{ccccc}-1&2&0&1&-1\\ -1/4&0&3/4&0&1/2\\ 7/4&0&-1/4&1&-1/2\\ -1/4&-1&1/4&0&3/2\\ 1/2&0&1/2&0&0\\ \end{array}\right]$
(C)

$\displaystyle\left[\begin{array}[]{ccccc}-1&-1/4&1/4&3/4&-1/2\\ 2&0&0&-1&1\\ 1&3/4&-7/4&-1/4&1/2\\ 0&0&1&0&0\\ -1&1/2&-3/2&1/2&0\\ \end{array}\right]$
(D)

$\displaystyle\left[\begin{array}[]{ccccc}-1&2&1&0&-1\\ -1/4&0&3/4&0&1/2\\ 1/4&0&-7/4&1&-3/2\\ 3/4&-1&-1/4&0&1/2\\ -1/2&1&1/2&0&0\\ \end{array}\right]$
(E)

$\displaystyle\left[\begin{array}[]{ccccc}-1&-1/4&1/4&3/4&-1/2\\ 2&0&0&-1&1\\ 1&3/4&-7/4&-1/4&1/2\\ 0&0&1&0&0\\ -1&1/2&-3/2&1/2&0\\ \end{array}\right]$

Workshop 1: Determinant of a matrix

Find the determinant of the $10\times 10$ matrix of the form:

\displaystyle\mathbf{V}=\left[\begin{array}[]{cccccc}1&-1&-2&0&\ldots&0\\ -1&1&-1&-2&\ddots&\vdots\\ -2&-1&\ddots&\ddots&\ddots&0\\ 0&-2&\ddots&\ddots&\ddots&-2\\ \vdots&\ddots&\ddots&-1&\ddots&-1\\ 0&\ldots&0&-2&-1&1\\ \end{array}\right]

Workshop 2: Inverse of a matrix

Find the inverse of the matrix $\mathbf{V}$ defined in the previous question. Find the (7,8) element of $\mathbf{V}^{-1}$ to 3 decimal places.

Workshop 3: Determinants of large matrix

Consider other matrices of the same general form as $\mathbf{V}$ . What size $n$ of $n\times n$ matrix leads to a matrix with determinant 190231?

Workshop 4: Largest determinant of a $3\times 3$ matrix

What is the largest possible determinant that a symmetric $3\times 3$ matrix, all of whose entries lie in the set of integers $\{1,2,3,4,5,6\}$ can have? Use R to determine the answer.