• The determinant is zero when the matrix has no inverse.
• The product of the pivots is the determinant
• The determinant changes sign when two rows (or two columns) are exchanged.

Basic properties

• The determinant of the n by n identity matrix is 1.
• The determinant changes sign when two rows are exchange.
• The determinant is a linear function of each row separately.
• If two rows of A are equal, then $det A = 0$.
• Subtracting a multiple of one row from another row leaves $det A$ unchanged.
• A matrix with a row of zeros has $det A = 0.$
• If $A$ is triangular then $det A = a_{11} a_{22}...a_{nn}$ = product of diagonal entries.
• If $A$ is singular then $det A = 0$. If A is invertible then $det A ≠0$.
• The determinant of $AB$ is $det A$ times $det B: |AB| = |A | |B|$.
• The transpose $A^T$ has the same determinant as $A$.