Determinant Basics

bmatrix a & b c & d bmatrix = ad - bc.

The idea

The determinant condenses a square matrix into a single number that records how the matrix scales area and whether it collapses the plane.

Draw the two columns of $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ as arrows from the origin, $(a, c)$ and $(b, d)$. The two arrows span a parallelogram. The determinant of $A$ is $\det A = ad - bc,$ and $|\det A|$ is the area of that parallelogram.

The sign carries its own information: $\det A$ is positive when the second column lies counterclockwise from the first, and negative when it lies clockwise. So the determinant records orientation as well as area.

The case $\det A = 0$ says the parallelogram has zero area, which happens exactly when the two columns lie along a single line. A zero determinant therefore detects parallel columns — the matrix flattens the plane onto a line.

Ways to work on it

• Walkthrough. We compute a 2 × 2 determinant together.
• Practice. Random 2 × 2 with positive integer entries.
• Hardest. One step. Entries can be negative — sign discipline matters.

Not sure where to start? Take the ten-question placement test.