Chapter 8.4, Problem 96E

a.

To determine

To Find: the determinant of four matrices in which the entries are consecutive integers using a graphing utility.

Explanation of Solution

Given:

The given example of matrix is

  $\left[\begin{array}{ccc}4& 5& 6\\ 7& 8& 9\\ 10& 11& 12\end{array}\right]$

Concept used:

Steps to use a graphing calculator for finding the determinant of a square matrix:

1. In the graphing device press “ $2^{\text{nd}}{x}^{-1}$ ” for entering the matrix
2. Choose “ EDIT” option for changing the order of the matrix
3. Put the entries of the matrix.
4. Quit the screen
5. Press “ $2^{\text{nd}}{x}^{-1}$ ” button again and select “MATH” option, then select “ $\det \left(\left|A\right|\right)$” and enter the matrix A for finding the determinant

Calculation:

Let’s consider for matrices as example

  $A=\left[\begin{array}{ccc}0& 1& 2\\ 3& 4& 5\\ 6& 7& 8\end{array}\right],B=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right],C=\left[\begin{array}{ccc}2& 3& 4\\ 5& 6& 7\\ 8& 9& 10\end{array}\right]$ and $D=\left[\begin{array}{ccc}3& 4& 5\\ 6& 7& 8\\ 9& 10& 11\end{array}\right]$

Example 1

  $A=\left[\begin{array}{ccc}0& 1& 2\\ 3& 4& 5\\ 6& 7& 8\end{array}\right]$

Use step 1 and 2 mentioned above

  

Quit the screen and do step 4 and 5

  

Hence determinant of A is 0,

Now example 2

  $B=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$

Use step 1 and 2 mentioned above

  

Quit the screen and do step 4 and 5

  

Hence determinant of B is 0

Example 3

Let $C=\left[\begin{array}{ccc}2& 3& 4\\ 5& 6& 7\\ 8& 9& 10\end{array}\right]$

Use step 1 and 2 mentioned above

  

Quit the screen and do step 4 and 5

  

Hence determinant of C is 0

Example 4

Let $D=\left[\begin{array}{ccc}3& 4& 5\\ 6& 7& 8\\ 9& 10& 11\end{array}\right]$

Use step 1 and 2 mentioned above

  

Quit the screen and do step 4 and 5

  

Hence determinant of D is 0

Here it can be observed that the determinant of a square matrix with consecutive integers as its entries is 0.

Conclusion:

Determinant of a square matrix with consecutive integers as its entries is 0.

b.

To determine

To verify: the conjecture of part (a).

Explanation of Solution

Given:

The given matrix is

  $\left[\begin{array}{ccc}4& 5& 6\\ 7& 8& 9\\ 10& 11& 12\end{array}\right]$

Conjecture: Determinant of a square matrix with consecutive integers as its entries is 0.

Concept used:

If the determinant of the given matrix is 0 then obtained conjecture is true.

Calculation:

  $\left[\begin{array}{ccc}4& 5& 6\\ 7& 8& 9\\ 10& 11& 12\end{array}\right]$

Now find its determinant as

  $\begin{array}{c}\left|\begin{array}{ccc}4& 5& 6\\ 7& 8& 9\\ 10& 11& 12\end{array}\right|=4\left(\left(8\times 12\right)-\left(9\times 11\right)\right)-5\left(\left(7\times 12\right)-\left(9\times 10\right)\right)+6\left(\left(7\times 11\right)-\left(8\times 10\right)\right)\\ =4\left(\left(96\right)-\left(99\right)\right)-5\left(\left(84\right)-\left(90\right)\right)+6\left(\left(77\right)-\left(80\right)\right)\\ =4\left(-3\right)-5\left(-6\right)+6\left(-3\right)\\ =-12+30-18\\ =0\end{array}$

Since the determinant of the matrix is 0, therefore the conjecture is verified.