Chapter EP, Problem 4.5.7EP

To determine

The solution of the determinant using diagonals.

Answer to Problem 4.5.7EP

  $-14$ .

Explanation of Solution

Given information:

Evaluate each determinant using diagonals.

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

Calculation:

The given matrix:

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

To find the determinant for the given matrix by using the following steps are as shown below:

Firstly write the first two column to the right of the determinant:

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

Secondly, find the products of the elements of the diagonals. First column is the elements from the left side of the determinant and the second column is the elements from the right side.

  $\begin{array}{l}6(5)(-2)=-60\text{ (}-\text{1)(5)(4)=}-\text{20}\\ 4(-8)(4)=-128\text{ 6}(-8)(3)\text{=}-\text{144}\\ (-1)(2)(3)=-6\text{ 4}(2)(-2)=-16\end{array}$

Now, find the sum of each group, the left side and the right side.

  $\begin{array}{l}-60+(-128)+(-6)-20+(-144)+(-16)\\ =-194\text{ =}-180\end{array}$

Subtract the result of the second group from the result of the first group.

  $\begin{array}{l}-194-(-180)\\ -194+180\\ -14\end{array}$

Therefore, the determinant of the matrix is $-14$ .