Problem-7

Question

What is the geometric interpretation of the determinant of a \(2 \times 2\) matrix?

Solution

The absolute value of a \(\displaystyle 2\times 2\) determinant gives the area of the parallelogram formed by the two column vectors. For example, consider the matrix: \[ A = \begin{bmatrix} 1 & 2\\ 2 & 1 \end{bmatrix} \] The two column vectors are plotted below and the parallelogram formed by them is completed. The area of this parallelogram is given by the absolute value of \(\text{det}(A)\).