Determinant of a Matrix: Formula, Rules, and Worked 2×2 and 3×3 Examples
The determinant of a matrix is a single number that tells you whether the matrix is invertible and how it scales area or volume. This guide shows the 2×2 and 3×3 formulas with worked examples and the rules that save time.
The determinant is a single number computed from a square matrix. For a 2×2 matrix it is
det = ad − bc: multiply down the main diagonal and subtract the product of the other diagonal. It tells you whether the matrix is invertible and how it scales area or volume.
Drop any matrix into our determinant calculator to get the answer instantly, including sizes beyond 3×3 where hand arithmetic gets painful.
The 2×2 determinant formula
Multiply down the main diagonal, then subtract the other product. For a generic two-by-two matrix the determinant is the top-left times bottom-right entry minus the top-right times bottom-left entry. That single line is the whole rule.
Work through a clean example. Take a matrix with no negatives so the subtraction is direct.
Watch the signs when an entry is negative. Subtracting a negative product turns into an addition, which is where careless errors creep in.
The 3×3 determinant by cofactor expansion
Expand along the top row. Each top-row entry multiplies the 2×2 determinant you get by crossing out its own row and column. The signs alternate plus, minus, plus across the row.
Work a full example end to end. Multiply each top-row entry by its 2×2 minor, apply the alternating signs, then add the three terms.
A determinant of zero means the matrix is singular. Its rows or columns are linearly dependent, so it has no inverse.
Expand along a row or column with zeros to save effort. Any term multiplied by zero drops out before you compute its minor.
Sarrus's rule for a faster 3×3
Copy the first two columns to the right, then sum diagonals. Add the three down-right products and subtract the three down-left products. It returns the same answer as cofactor expansion, only laid out visually.
Sarrus's rule does not extend to 4×4 or larger matrices, which is where most students get burned applying it out of habit. For anything beyond 3×3, lean on the determinant calculator.
What the determinant actually tells you
The value carries meaning beyond the arithmetic. A zero result flags a singular matrix, a nonzero result confirms an inverse exists, the absolute value gives the area or volume scale factor, and the sign records whether orientation is preserved or flipped.
det = 0 → singular, no inverse, rows/columns dependent
det ≠ 0 → invertible, inverse exists
|det| → area (2D) or volume (3D) scale factor
sign(det) → + preserves orientation, − flips itPicture a 2×2 matrix acting on the plane. A positive determinant stretches every area by that factor, while a negative one stretches by the absolute value and reflects the plane.
Rules that save work
A handful of properties let you skip most of the arithmetic. Triangular matrices and rows of zeros collapse instantly, and row operations either change the determinant in a known way or leave it alone.
triangular matrix → product of the diagonal entries row of zeros → determinant is 0 two identical rows → determinant is 0 swap two rows → determinant changes sign multiply a row by k → determinant multiplies by k add a multiple of one row → determinant unchanged
That last property is what makes a hand-computed 4×4 determinant feasible. Use row operations to create zeros, then expand along the row holding the most of them.
How the determinant connects to the rest of linear algebra
The determinant threads through nearly every other tool in the subject. It controls the inverse formula, drives Cramer's rule, defines eigenvalues through the characteristic equation, and compactly encodes a cross product.
Cramer's rule solves linear systems using ratios of determinants, and the cross product of two 3D vectors is written as a single 3×3 determinant. For the companion operation, see our dot product guide.
Common mistakes to avoid
Most errors come from signs and from misusing shortcuts. The middle cofactor term is subtracted, not added. Sarrus's rule applies only to 3×3 matrices. And the determinant is a specific combination of products, not the trace, which is just the sum of the diagonal.
The determinant compresses a whole matrix into one number that decides invertibility, scaling, and orientation. Master the 2×2 and 3×3 formulas by hand, then reach for the determinant calculator on anything larger. Related reading: the dot product of two vectors and the inner product guide.
Common questions
A zero determinant means the matrix is singular and has no inverse. Its rows (or columns) are linearly dependent, so the matrix collapses space rather than scaling it cleanly. Any linear system built on it lacks a unique solution.
For a 2×2 matrix the determinant is det = ad − bc. Multiply the main-diagonal entries (top-left and bottom-right), then subtract the product of the off-diagonal entries (top-right and bottom-left).
No. Sarrus's rule is a visual shortcut that only works for 3×3 matrices. For 4×4 and larger, use cofactor expansion, ideally after row operations create zeros, or let a calculator handle the arithmetic.
The trace is the sum of the diagonal entries. The determinant is a specific signed combination of products across the whole matrix. They measure different things and are easy to confuse on small matrices.
