Chapter 4.5, Problem 6CYU

To determine

To find: The determinants using diagonals.

Answer to Problem 6CYU

The determinant of the matrix using diagonals is -24.

Explanation of Solution

Given,

The determinant is given as,

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

Calculation:

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

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

Calculate the diagonal products.

  $\begin{array}{l}2\left(6\right)\left(-6\right)=-72\\ -3\left(-2\right)\left(4\right)=24\\ 5\left(-4\right)\left(-1\right)=20\end{array}$

Calculate the sum of the products.

  $-72+24+20=28$

Similarly, calculate the diagonal products from the other side

  $\begin{array}{l}4\left(6\right)\left(5\right)=120\\ -1\left(-2\right)\left(2\right)=4\\ -6\left(-4\right)\left(-3\right)=-72\end{array}$

Calculate the sum of the products.

  $120+4+\left(-72\right)=52$

Calculate the determinant of the matrix.

  $28-52=-24$

Thus, the determinant of the matrix using diagonals is -24.