Q4. - Algebra 2

Question

Find a counterexample to disprove the following statement.

Two different matrices can never have the same determinant.

Step-by-Step Solution

Answer

The matrix $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ and $B\left[\begin{array}{cc}8& 3\\ 5& 2\end{array}\right]$disprove the statement “Two different matrices can never have the same determinant.”.

Step 1 - Assume the matrices

In order to disprove the statement “Two different matrices can never have the same determinant” consider two $2\times 2$ matrices $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ and $B\left[\begin{array}{cc}8& 3\\ 5& 2\end{array}\right]$ and calculate there respective determinants.

Step 2 -Define a determinant of second order matrix

The determinant of second order matrix is found by calculating the difference of the product of the two diagonals, that is.,  $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$.

Step 3 - Find the determinants

Calculate the determinants of matrix A and B and observe if they provide the same value.

$\begin{array}{c}\left|A\right|=\left|\begin{array}{c}\begin{array}{cc}1& 0\\ 0& 1\end{array}\end{array}\right|\\ =1\times \left(1\right)-0\left(0\right)\\ =1\end{array}$

And, 

$\begin{array}{c}\left|B\right|=\left|\begin{array}{c}\begin{array}{cc}8& 3\\ 5& 2\end{array}\end{array}\right|\\ =8\times \left(2\right)-3\left(5\right)\\ =16-15\\ =1\end{array}$ 

Clearly, A and B are two different matrices since their corresponding values are different and also they have same determinants which disprove the statement that “Two different matrices can never have the same determinant”.