# Determinants For Class 12

Every square matrix A of the order n, can associate a number called determinants of the square matrix A.

Determinant of the order 1×1-

Consider a matrix A = [a], then the determinant of the matrix is equal to a.

Determinant of the order 2×2-

If the order of the matrix is 2, then the determinants is defined matrix A, where A is-

$A= \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}$

Determinant of A = $\left |A \right |= \begin{vmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{vmatrix}$ $= a_{11}. a_{22} – a_{21}.a_{12}$

Similarly we can find the determinant of the order 3×3

Let us now look on to the properties of the Determinants:

Property 1- The value of the determinant remains unchanged if the rows and columns of a determinant are interchanged.

Property 2- If any two rows (or columns) of a determinants are interchanged, then sign of determinants changes.

Property 3- If any two rows or columns of a determinant are equal or identical, then the value of the determinant is 0.

Property 4- If each element of a row or a column is multiplied by a constant value k, then the value of the determinant originally obtained is multiplied with k.