Imagine you have a 2x2 grid or matrix where:
- The top-left cell has the number 'a'
- The top-right cell has the number 'b'
- The bottom-left cell has the number 'c'
- The bottom-right cell has the number 'd'
Now, think of this grid's columns as arrows or directions on a flat surface. The first column points in the direction of (a,c) and the second column points in the direction of (b,d).
These two directions outline a shape on the flat surface, which looks like a slanted rectangle - a parallelogram. The space enclosed by this shape is its area.
The formula 'ad - bc' gives us a number representing this area. If we take the absolute value (ignore the negative sign if any) of this result, we get the exact area of the parallelogram. If the result is negative, it's like looking at the shape from the other side, but the size or area remains the same.
In simple words, if you draw these directions on paper and sketch the shape they make, the space they enclose is equal to the absolute value of 'ad - bc' from our grid.
Imagine you have two vectors starting from the origin in a 2D plane. These vectors define a parallelogram when combined with their respective parallel counterparts.
The determinant of the matrix that contains these vectors as its columns gives us the area of the parallelogram they span.
1. If the two vectors are on the same line (or one is a scalar multiple of the other), they don't really span a 2D space; they span a line. The "parallelogram" they define collapses into a line, and its area is 0. This is analogous to the determinant being 0, indicating that the matrix does not have a unique inverse.
2. If the two vectors are orthogonal (i.e., at 90 degrees to each other), the parallelogram becomes a rectangle. The area of this rectangle (base x height) directly corresponds to the determinant's magnitude.
3. In all other cases, the vectors span a slanted parallelogram, and its area is given by the determinant's absolute value. The sign of the determinant tells us about the "orientation" of the matrix.
In finance, let's imagine these vectors represent two portfolios and their expected returns over two different time intervals. If the determinant of this matrix is zero, it means they are perfectly correlated and don't offer diversification. The "area" they span (parallelogram) collapses, showing no diversification benefit.
The graph below shows the difference between the two areas of the cross product (green and grey rectangles) and displays their respective values. The determinant (area of the parallelogram) is represented by the net difference between these areas.
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