If we interchange any two rows (or columns), then sign of determinant changes. If any two rows or any two columns are identical or proportional, then value of determinant is zero.
If we multiply each element of a row or a column of a determinant by constant k, then value of determinant is multiplied by k.
Multiplying a determinant by k means multiply elements of only one row (or one column) by k.
If elements of a row or a column in a determinant can be expressed as sum of two or more elements, then the given determinant can be expressed as sum of two or more determinants.
If to each element of a row or a column of a determinant the equi-multiples of corresponding elements of other rows or columns are added, then value of determinant remains same.
A square matrix A has inverse if and only if A is non-singular.
Unique solution of equation AX = B is given by X = A-1B, where A ≠ 0.
A system of equation is consistent or inconsistent according as its solution exists or not.
For a square matrix A in matrix equation AX = B
- |A| ≠ 0, there exists unique solution
- |A| = 0 and (adj A) B ≠ 0, then there exists no solution
- |A| = 0 and (adj A) B = 0, then system may or may not be consistent.