Chapter 16.8, Problem 9WE

To determine

To calculate: The value of the determinant,

  $\left|\begin{array}{ccc}5& 1& 1\\ 1& 5& 1\\ 1& 1& 5\end{array}\right|$

Answer to Problem 9WE

The value of the determinant is $112$ .

Explanation of Solution

Given information:

The determinant:

  $\left|\begin{array}{ccc}5& 1& 1\\ 1& 5& 1\\ 1& 1& 5\end{array}\right|$

Formula used:

For the matrix of order $2\times 2$ ,

  $A=\left[\begin{array}{ll}p\hfill & q\hfill \\ r\hfill & s\hfill \end{array}\right]$

The determinant is given as $\left|A\right|=ps-qr$ .

Calculation:

Consider the determinant

  $\left|\begin{array}{ccc}5& 1& 1\\ 1& 5& 1\\ 1& 1& 5\end{array}\right|$

Recall that for the matrix of order $2\times 2$ ,

  $A=\left[\begin{array}{ll}p\hfill & q\hfill \\ r\hfill & s\hfill \end{array}\right]$

The determinant is given as $\left|A\right|=ps-qr$ .

Expand by minors about first row,

  $\begin{array}{c}\left|\begin{array}{ccc}5& 1& 1\\ 1& 5& 1\\ 1& 1& 5\end{array}\right|=5\left(25-1\right)-1\left(5-1\right)+1\left(1-5\right)\\ =120-4-4\\ =112\end{array}$

Thus, the value of the determinant is $112$ .