Q6. - Algebra 2

Question

Show that the value of $\left|\begin{array}{ccc}-2& 3& 5\\ 0& -1& 4\\ 9& 7& 2\end{array}\right|$ is the same whether you use expansion by minors or diagonals.

Step-by-Step Solution

Answer

The value of the determinant $\left|\begin{array}{ccc}-2& 3& 5\\ 0& -1& 4\\ 9& 7& 2\end{array}\right|=213$ is same whether you use expansion by minors or diagonals.

Step 1 - Calculate determinant using expansion by minors

Use first row for the expansion by minors.

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

 

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

Apply this definition to find the $2\times 2$determinant.

$\begin{array}{c}\left|\begin{array}{c}\begin{array}{ccc}-2& 3& 5\\ 0& -1& 4\\ 9& 7& 2\end{array}\end{array}\right|=-2\left|\begin{array}{c}\begin{array}{cc}-1& 4\\ 7& 2\end{array}\end{array}\right|-3\left|\begin{array}{c}\begin{array}{cc}0& 4\\ 9& 2\end{array}\end{array}\right|+5\left|\begin{array}{c}\begin{array}{cc}0& -1\\ 9& 7\end{array}\end{array}\right|\\ =-2(2(-1)-4\left(7\right))-3(0-9\left(4\right))+5(0-9(-1))\\ =-2(-2-28)-3(-36)+5\left(9\right)\\ =60+108+45\\ =213\end{array}$

Step 2 - Calculate determinant using diagonals

In method of calculating determinant by using diagonals first two columns are rewritten outside to the right side of the determinant.

 

$\left|\begin{array}{ccc}-2& 3& 5\\ 0& -1& 4\\ 9& 7& 2\end{array}\right|\begin{array}{c}-2\\ 0\\ 9\end{array} \begin{array}{c}3\\ -1\\ 7\end{array}$

 

The products are calculated diagonally in two ways – bottom products and top products. For bottom product, draw diagonals from each element of the top row of the determinant downward to the right. Find the product of the elements of each respective diagonal.

For top product, draw diagonals from each element of the bottom row of the determinant upward to the right. Find the product of the elements of each respective diagonal.

Step 3 - Add and subtract the products and state conclusion

In order to find the determinant, add the bottom products and subtract the top products.

 

$\begin{array}{c}4+108+0-(-45)-(-56)-0=112+45+56\\ =213\end{array}$

 

Therefore, the value of the determinant is same whether you use expansion by minors or diagonals.