Final Answer:
The determinant of the matrix [[0, -3], [7, 0]] is 21.
Step-by-step explanation:
The determinant of a 2x2 matrix [ [a, b], [c, d] ] is given by the formula ad - bc. In our case, a = 0, b = -3, c = 7, and d = 0. So, the determinant is (0 * 0) - (-3 * 7) = 21.
In more detail, the determinant represents the scaling factor by which the matrix transforms the area or volume of a geometric shape. For a 2x2 matrix, it corresponds to the signed area of the parallelogram formed by the column vectors. In our matrix, the columns [0, 7] and [-3, 0] define a parallelogram with base length 7 and height 3. The negative sign in front of the -3 in the matrix leads to a counterclockwise orientation, resulting in a positive determinant. The magnitude of 21 indicates the scaling factor, meaning the parallelogram's area is 21 times larger than the unit square. Therefore, the determinant, 21, encapsulates both the orientation and the scaling effect of the matrix on the geometric space.