Suppose $J=\left[\begin{array}{cc}-2 & 6 \\ 4 & 8\end{array}\right]$.
What is the value of $|J|$?
We are given a \(2 \times 2\) matrix \(J\) defined as:
\[J = \begin{bmatrix} -2 & 6 \ 4 & 8 \end{bmatrix}\]
To find the determinant of a \(2 \times 2\) matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), we use the standard determinant formula:
\[|A| = ad - bc\]
In this case, the matrix elements are:
\(a = -2\)
\(b = 6\)
\(c = 4\)
\(d = 8\)
Our objective is to compute the value of the determinant \(|J|\).
Using the values identified in the previous step, we substitute them into the determinant formula:
\[|J| = (-2) \times (8) - (6) \times (4)\]
Now, we perform the arithmetic operations:
\[(-2) \times (8) = -16\]
\[(6) \times (4) = 24\]
Subtracting these two products gives:
\[|J| = -16 - 24 = -40\]
Therefore, the value of the determinant of matrix \(J\) is \(-40\).
In physics, the determinant of a matrix can represent the scaling factor of the transformation of an area or volume. For instance, if you are working with a coordinate transformation in engineering, the determinant of the transformation matrix tells you how the area of a shape changes after the transformation. A negative determinant, like the one we calculated, further indicates that the transformation includes a reflection, meaning the orientation of the shape is reversed.