Suppose $J=\left[\begin{array}{cc}-2 & 6 \\ 4 & 8\end{array}\right]$.
What is the value of $|J|$?
The given matrix \(J\) is a \(2 \times 2\) matrix defined as:
\[J = \begin{bmatrix} -2 & 6 \ 4 & 8 \end{bmatrix}\]
To find the determinant of a \(2 \times 2\) matrix in the form \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), we use the standard formula:
\[|A| = ad - bc\]
In our specific matrix \(J\), the elements are:
\(a = -2\)
\(b = 6\)
\(c = 4\)
\(d = 8\)
These values correspond to the position of each element in the matrix structure.
Now, substitute the values \(a = -2\), \(b = 6\), \(c = 4\), and \(d = 8\) into the determinant formula:
\[|J| = (-2)(8) - (6)(4)\]
First, perform the multiplications:
\[(-2)(8) = -16\]
\[(6)(4) = 24\]
Now, substitute these products back into the determinant expression:
\[|J| = -16 - 24\]
Finally, perform the subtraction:
\[|J| = -40\]
Therefore, the determinant of the matrix \(J\) is \(-40\).
In data science and engineering, matrices are used to represent linear transformations. The determinant is a critical value that tells us if a transformation scales space (changes its area or volume) or if it collapses it. For example, if you are working with a 2D graphics engine and want to rotate or scale an image, the determinant helps determine if the image is being flipped (if the determinant is negative) or resized. Understanding how to calculate it allows you to analyze these transformations effectively.