Chapter 4.5, Problem 9CYU

To determine

To find: The determinants using diagonals.

Answer to Problem 9CYU

The determinant of the matrix using diagonals is 72.

Explanation of Solution

Given,

The determinant is given as,

  $\left|\begin{array}{ccc}8& 3& 4\\ 2& 4& 2\\ 1& 6& 5\end{array}\right|$

Calculation:

Write the first two columns to the right side of the determinant.

  $\left|\begin{array}{ccc}8& 3& 4\\ 2& 4& 2\\ 1& 6& 5\end{array}\right|\begin{array}{cc}8& 3\\ 2& 4\\ 1& 6\end{array}$

Calculate the diagonal products.

  $\begin{array}{l}8\left(4\right)\left(5\right)=160\\ 3\left(2\right)\left(1\right)=6\\ 4\left(2\right)\left(6\right)=48\end{array}$

Calculate the sum of the products.

  $160+6+48=214$

Similarly, calculate the diagonal products from the other side

  $\begin{array}{l}1\left(4\right)\left(4\right)=16\\ 6\left(2\right)\left(8\right)=96\\ 5\left(2\right)\left(3\right)=30\end{array}$

Calculate the sum of the products.

  $16+96+30=142$

Calculate the determinant of the matrix.

  $214-142=72$

Thus, the determinant of the matrix using diagonals is 72.