Understanding the Determinant Geometrically

What Is a Determinant?

A determinant is a scalar value associated with a square matrix. It contains important information about matrix behavior for transformations, volume scaling, and orientation preservation.
If you’re in high school, you probably know it as ab-cd for a 2D square matrix A

But what does ab-cd actually mean? and why the det is calculated as ab — cd?

Geometric Interpretations

Consider the following 2 x 2 matrix A

This A is the Augmentation of two vectors

When we plot these vectors in a graph,

Now, consider a parallelogram formed by these two vectors:

The area of this parallelogram can be calculated by 4 * 2 = 8 which is the det of the matrix. Hence, a determinant of a 2D matrix gives the area of the parallelogram formed by the two vectors.

This is easy because b and c are zeroes in our example and the vectors are orthogonal. Lets say we do not know the values of a,b,c and d . Consider matrix A

Matrix A is augmentation of column vectors B and C

When we plot on a graph,

We then draw parallels for the two vectors, they meet at (a+b, c+d)

We then consider the rectangle formed by the points (a+b, c+d), (a+b, 0), (0, c+d) and origin

As seen from the image, the area of parallelogram is the area of bigger rectangle i.e. (a+b)*(c+d) minus the area in the rectangle that is not occupied by the parallelogram .

The area in rectangle that is not occupied by parallelogram can be known with help of the coordinates of the parallelogram.

Area in rectangle that is not occupied by parallelogram is bc + bc+ ac/2 + ac/2 + bd/2 + bd/2.

Hence, the area of parallelogram is,

(a+b)*(c+d) — ( bc + bc+ ac/2 + ac/2 + bd/2 + bd/2) where (a+b)*(c+d) is area of bigger rectangle

By applying distributive law, the equation is simplified to ac + ad + bc + bd — (bc+ bc+ ac + bd)

Which is same as,

ac + ad + bc + bd — bc — bc — ac — bd

We rearrange them ,

ac — ac + ad + bc — bc — bc + bd — bd

The ac , bc and bd pair gets cancelled leaving ad-bc which is the determinant of a 2D matrix!

When the det is negative, it simply means that the direction of the parallelogram is clockwise.

If two vectors are collinear or parallel, we cannot form a parallelogram. So det is zero.

3D Case: Volume

The same intuition can be generalized to other dimensions. In 3D, det gives the volume of the parallelogram formed by the 3 column vectors of a 3 X 3 matrix.

• When the det is positive, the parallelepiped has a right-handed orientation, and the volume is positive.
• If negative, the parallelepiped has a left-handed orientation, and the volume is negative.
• When the vectors are linearly dependent, the parallelepiped collapses to a plane. So, no volume and det becomes zero

1D Case: Length

In 1D we have one vector:

The determinant of this “matrix”, which is the element itself, is simply the magnitude or length of the vector

This tells us how far the vector extends in its direction.

Feel free to post your queries, critics, and ideas :)