The determinant turns a whole matrix into a single number. For a matrix the recipe is short — two products and a subtraction — but that subtraction is where most mistakes happen.

Condition Is it square?

A determinant only exists for a square matrix — one with the same number of rows as columns. Always check this first.

2 rows, 2 columns — square

2 rows, 3 columns — not square

The Rule Main diagonal, minus the other diagonal

Multiply along the main diagonal, then subtract the product of the other diagonal.

The minus sign is not optional — forgetting it is the single most common mistake in this calculation.

Example 1

— main diagonal

— other diagonal

The determinant is always a single number: here, .

Example 2 Watch the minus sign

Replace the with : .

— main diagonal, unchanged

— other diagonal

Subtracting a negative flips into addition. The same two entries, one sign changed, and the determinant jumps from to .

Summary

A determinant only exists for a square matrix — rows must equal columns.
For , the determinant is .
.
Flip one entry’s sign and the subtraction becomes addition: .

Determinant of a Matrix