Chapter 4.5, Problem 7CYU

To determine

To find: The determinants using diagonals.

Answer to Problem 7CYU

The determinant of the matrix using diagonals is -284.

Explanation of Solution

Given,

The determinant is given as,

  $\left|\begin{array}{ccc}8& 4& 0\\ -2& -6& -1\\ 5& -3& 6\end{array}\right|$

Calculation:

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

  $\left|\begin{array}{ccc}8& 4& 0\\ -2& -6& -1\\ 5& -3& 6\end{array}\right|\begin{array}{cc}8& 4\\ -2& -6\\ 5& -3\end{array}$

Calculate the diagonal products.

  $\begin{array}{l}8\left(6\right)\left(-6\right)=-288\\ 4\left(-1\right)\left(5\right)=-20\\ 0\left(-2\right)\left(-3\right)=0\end{array}$

Calculate the sum of the products.

  $-288+\left(-20\right)=-308$

Similarly, calculate the diagonal products from the other side

  $\begin{array}{l}5\left(-6\right)\left(0\right)=0\\ -3\left(-1\right)\left(8\right)=24\\ 6\left(-2\right)\left(4\right)=-48\end{array}$

Calculate the sum of the products.

  $0+24+\left(-48\right)=-24$

Calculate the determinant of the matrix.

  $-308+24=-284$

Thus, the determinant of the matrix using diagonals is -284.