Question

find the value of the determinant.
\

$$\begin{vmatrix}9 & -2 & -3\\\\8 & 0 & -1\\\\7 & 3 & 7\\end{vmatrix}$$

the determinant value is \boxed{}

Explanation:

Step1: Recall the formula for a 3x3 determinant

For a matrix \(

$$\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}$$

\), the determinant is \(a(ei - fh)-b(di - fg)+c(dh - eg)\).

Step2: Identify the values

Here, \(a = 9\), \(b=-2\), \(c = -3\), \(d = 8\), \(e = 0\), \(f=-1\), \(g = 7\), \(h = 3\), \(i = 7\).

Step3: Calculate each part

First part: \(a(ei - fh)=9(0\times7-(-1)\times3)=9(0 + 3)=27\)
Second part: \(-b(di - fg)=-(-2)(8\times7-(-1)\times7)=2(56 + 7)=2\times63 = 126\)
Third part: \(c(dh - eg)=-3(8\times3-0\times7)=-3(24 - 0)=-72\)

Step4: Sum the parts

Add the three parts: \(27+126-72=81\)

Answer:

\(81\)