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Problem Statement

The problem C23, selected page 353,
Doing the computations by hand, find the determinant of the matrix below.

##      [,1] [,2] [,3]
## [1,]    1    3    2
## [2,]    4    1    3
## [3,]    1    0    1

Solution

Determinant of the matrix above can be compute by hand as following

det (matr)

= \(\begin{vmatrix} 1 & 3 & 2 \\ 4 & 1 & 3 \\ 1 & 0 & 1 \end{vmatrix}\)

= \(1\begin{vmatrix} 1 & 3 \\ 0 & 1 \end{vmatrix}-3\begin{vmatrix} 4 & 3 \\ 1 & 1 \end{vmatrix}+2\begin{vmatrix} 4 & 1 \\ 1 & 0 \end{vmatrix}\)

= 1(1X1 - 3X0) - 3(4X1 - 3X1) + 2(4X0 -1X1)
= 1(1) - 3(1) + 2(-1)
= 1 - 3 - 2
= -4

Above we used
\(Determinant\quad of\quad Matrices\quad of\quad Size\quad Two\\ suppose\quad that\quad A\quad =\quad \begin{vmatrix} a & b \\ c & d \end{vmatrix}\quad then\quad det(A)\quad =\quad ad\quad -\quad bc\)

We can verify the result using R

## [1] -4