A determinant assigns one number to a square matrix. That number reveals whether the associated linear transformation preserves a nonzero area or volume and whether the matrix has an inverse. A determinant equal to zero means that dimensions collapse: rows or columns are linearly dependent, and a system with that coefficient matrix cannot have one unique solution.
The 2 × 2 calculation and its geometric sign
For a 2 by 2 matrix, multiply the entries on the main diagonal and subtract the product on the other diagonal. The absolute value is the area scale factor of the transformation. The sign records orientation: a negative determinant includes a reflection, while a positive one preserves orientation.
Use the controls to see both the arithmetic and the parallelogram formed by the matrix's column vectors. When the two columns fall on the same line, the parallelogram has zero area and the determinant becomes zero.
Explore a 2 × 2 determinant
The shaded parallelogram has area equal to the determinant's absolute value.
Sarrus' rule works only for 3 × 3 matrices
For a 3 by 3 determinant, Sarrus' rule copies the first two columns to the right. Add the products along the three downward diagonals, then subtract the products along the three upward diagonals. The layout is a memory aid for six terms, not a general determinant algorithm.
Applying the same diagonal trick to a 4 by 4 matrix gives a wrong result. For larger orders, use row reduction, cofactor expansion, or a numerical factorization. Even at order three, write the positive and negative groups separately so a negative entry does not make the signs ambiguous.
Laplace expansion uses minors and cofactors
Delete row (i) and column (j) to obtain the minor associated with entry (a_{ij}). Its cofactor adds the alternating sign pattern that begins with plus in the upper-left corner. A Laplace expansion multiplies every entry of one chosen row or column by its cofactor and adds the results.
Any row or column gives the same determinant. Choose one containing several zeros to reduce the work. Laplace expansion is conceptually general, although it becomes inefficient for large dense matrices because the number of smaller determinants grows quickly.
Worked 3 × 3 determinant
Consider the matrix with rows (1,2,3), (0,4,5), and (1,0,6). Expanding along the second row is attractive because its first entry is zero. The two remaining cofactors produce the same value that Sarrus' rule gives.
The determinant is 22, so the matrix is invertible. A sign check is useful here: if two rows were swapped before calculating, the answer would be −22. Multiplying one row by 3 would multiply the determinant by 3, while adding a multiple of one row to another would leave it unchanged.
Fast checks and common limitations
Triangular matrices need no expansion: their determinant is the product of the main-diagonal entries. Identical or proportional rows force a zero determinant. The determinant of a product is the product of the determinants, and transposing does not change it. These properties make row operations an efficient calculation route.
A determinant exists only for a square matrix. Its value also depends on units and scale, so a small nonzero determinant does not automatically prove a numerical problem is unstable. For the surrounding vocabulary of order, entries and products, see the matrix overview; for the connection with invertibility, see inverse matrices.
