What is the volume of the parallelepiped defined by the columns of matrix \( \left[\begin{array}{ccc}0 & 3 & 6 \\ -6 & 9 & 5 \\ 0 & 0 & -9\end{array}\right] \) Volume \( = \) Ex. 1

To find the volume of the parallelepiped defined by the columns of the matrix \( \left[\begin{array}{ccc}0 & 3 & 6 \\ -6 & 9 & 5 \\ 0 & 0 & -9\end{array}\right] \), you can use the absolute value of the determinant of the matrix. First, let's calculate the determinant using the formula for a 3x3 matrix: \[ \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \] For our matrix \( A = \left[\begin{array}{ccc}0 & 3 & 6 \\ -6 & 9 & 5 \\ 0 & 0 & -9\end{array}\right] \): 1. \( a=0, b=3, c=6 \) 2. \( d=-6, e=9, f=5 \) 3. \( g=0, h=0, i=-9 \) Plugging these values into the formula gives: \[ \text{det}(A) = 0(9 \cdot -9 - 5 \cdot 0) - 3(-6 \cdot -9 - 5 \cdot 0) + 6(-6 \cdot 0 - 9 \cdot 0) \] \[ = 0 - 3(54) + 0 = -162 \] Thus, the volume of the parallelepiped is given by the absolute value of the determinant: \[ \text{Volume} = | -162 | = 162 \] So, the final volume \( = 162 \). For a quick way to visualize this, think of the parallelepiped as a 3D shape extending from one corner of a rectangular box, defined by the vectors represented by the columns of this matrix. When you determine the volume, you find out how much space this shape takes up in that box! Another fun fact is that the concept of determinants is used in several fields! For instance, in physics, determinants can help solve systems of equations, determine rotational effects in mechanics, and are essential in computer graphics for transformations like scaling and rotating shapes. It's like a behind-the-scenes wizard that makes 3D world-building work seamlessly!