Chapter 4.5, Problem 10CYU

To determine

To find: The determinants using diagonals.

Answer to Problem 10CYU

The determinant of the matrix using diagonals is 139.

Explanation of Solution

Given,

The determinant is given as,

  $\left|\begin{array}{ccc}-4& 3& 0\\ 1& 5& -2\\ -1& -8& -3\end{array}\right|$

Calculation:

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

  $\left|\begin{array}{ccc}-4& 3& 0\\ 1& 5& -2\\ -1& -8& -3\end{array}\right|\begin{array}{cc}-4& 3\\ 1& 5\\ -1& -8\end{array}$

Calculate the diagonal products.

  $\begin{array}{l}-4\left(5\right)\left(-3\right)=60\\ 3\left(-2\right)\left(-1\right)=6\\ 0\left(1\right)\left(8\right)=0\end{array}$

Calculate the sum of the products.

  $60+6+0=66$

Similarly, calculate the diagonal products from the other side

  $\begin{array}{l}-1\left(5\right)\left(0\right)=0\\ -8\left(-2\right)\left(-4\right)=-64\\ -3\left(1\right)\left(3\right)=-9\end{array}$

Calculate the sum of the products.

  $0-64-9=-73$

Calculate the determinant of the matrix.

  $66+73=139$

Thus, the determinant of the matrix using diagonals is 139.