Chapter 7.6, Problem 44E

a.

To determine

To calculate: the determinant of four matrices and make a conjecture based on the result

Answer to Problem 44E

The determinant of a matrix whose elements are integer consecutive integers is zero.

Explanation of Solution

Given:

Consider square matrices in which the entries are consecutive integers. An example of such a 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:

Example 1: Considering a matrix A and finding its determinant $\left|A\right|$

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

Using graphing calculator with the steps mentioned above

After step 1 and step 2

  

Quit the screen and do step 4 and step 5

  

Hence by graphing calculator, determinant of matrix A is 0

Or

  $\begin{array}{l}A=\left[\begin{array}{ccc}0& 1& 2\\ 3& 4& 5\\ 6& 7& 8\end{array}\right]\\ \left|A\right|=0\left|\begin{array}{cc}4& 5\\ 7& 8\end{array}\right|-1\left|\begin{array}{cc}3& 5\\ 6& 8\end{array}\right|+2\left|\begin{array}{cc}3& 4\\ 6& 7\end{array}\right|\\ \left|A\right|=0-1\left(24-30\right)+2\left(21-24\right)\\ \left|A\right|=0\end{array}$

Example 2: Considering a matrix B and finding its determinant $\left|B\right|$

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

Using graphing calculator with the steps mentioned above

After step 1and step 2

  

Quit the screen and do step 4 and step 5

  

Hence by graphing calculator, determinant of matrix B is 0

Or

  $\begin{array}{l}B=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]\\ B=\left|\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right|\\ \left|B\right|=1\left|\begin{array}{cc}5& 6\\ 8& 9\end{array}\right|-2\left|\begin{array}{cc}4& 6\\ 7& 9\end{array}\right|+1\left|\begin{array}{cc}4& 5\\ 7& 8\end{array}\right|\\ \left|B\right|=0\end{array}$

Example 3: Considering a matrix C and finding its determinant $\left|C\right|$

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

Using graphing calculator with the steps mentioned above

After step 1and step 2

  

Quit the screen and do step 4 and step 5

  

Hence by graphing calculator, determinant of matrix C is 0

Or

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

Expand along row 1

  $\begin{array}{l}\left|C\right|=\left|\begin{array}{ccc}2& 3& 4\\ 5& 6& 7\\ 8& 9& 10\end{array}\right|\\ \left|C\right|=2\left|\begin{array}{cc}6& 7\\ 9& 10\end{array}\right|-3\left|\begin{array}{cc}5& 7\\ 8& 10\end{array}\right|+4\left|\begin{array}{cc}5& 6\\ 8& 10\end{array}\right|\\ \left|C\right|=0\end{array}$

Example 4: Considering a matrix D and finding its determinant $\left|D\right|$

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

Using graphing calculator with the steps mentioned above

After step 1and step 2

  

Quit the screen and do step 4 and step 5

  

Hence by graphing calculator, determinant of matrix C is 0

Or

  $\begin{array}{l}D=\left[\begin{array}{ccc}3& 4& 5\\ 6& 7& 8\\ 9& 10& 10\end{array}\right]\\ \left|D\right|=3\left|\begin{array}{cc}7& 8\\ 10& 10\end{array}\right|-4\left|\begin{array}{cc}6& 8\\ 9& 10\end{array}\right|+5\left|\begin{array}{cc}6& 7\\ 9& 10\end{array}\right|\\ \left|D\right|=0\end{array}$

From above 4 examples we can conclude that the determinant of a matrix whose elements are integer consecutive integers is zero.

b.

To determine

To Verify: the conjecture.

Explanation of Solution

Given:

Consider square matrices in which the entries are consecutive integers. An example of such a matrix is

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

The determinant of a matrix whose elements are integer consecutive integers is zero.

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:

Consider square matrices in which the entries are consecutive integers.

Let a be an integer then the matrix formed with the consecutive integers will be $A=\left[\begin{array}{ccc}a& a+1& a+2\\ a+3& a+4& a+5\\ a+6& a+7& a+8\end{array}\right]$

Applying $R_2\to R_2-R_1$ and $R_3\to R_3-R_1$ , we get

  $A=\left[\begin{array}{ccc}a& a+1& a+2\\ 3& 3& 3\\ 6& 6& 6\end{array}\right]$

Now apply $R_3\to R_3-2R_2$

  $A=\left[\begin{array}{ccc}a& a+1& a+2\\ 3& 3& 3\\ 0& 0& 0\end{array}\right]$

If any row or column of a matrix is zero then its determinant is zero.

Hence $\left|A\right|=0$