Chapter 4.5, Problem 8CYU

To determine

To find: The determinants using diagonals.

Answer to Problem 8CYU

The determinant of the matrix using diagonals is 308.

Explanation of Solution

Given,

The determinant is given as,

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

Calculation:

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

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

Calculate the diagonal products.

  $\begin{array}{l}-5\left(-4\right)\left(4\right)=80\\ -3\left(-3\right)\left(8\right)=78\\ 4\left(-2\right)\left(-2\right)=16\end{array}$

Calculate the sum of the products.

  $80+\left(78\right)+16=174$

Similarly, calculate the diagonal products from the other side

  $\begin{array}{l}8\left(-4\right)\left(4\right)=-128\\ -2\left(-3\right)\left(-5\right)=-30\\ 4\left(-2\right)\left(-3\right)=24\end{array}$

Calculate the sum of the products.

  $-128+\left(-30\right)+\left(24\right)=-134$

Calculate the determinant of the matrix.

  $174-\left(-134\right)=308$

Thus, the determinant of the matrix using diagonals is 308.