Odisha State Board CHSE Odisha Class 12 Math Notes Chapter 5 Determinants will enable students to study smartly.

## CHSE Odisha 12th Class Math Notes Chapter 5 Determinants

Evaluation of determinants:

Note: We can evaluate a determinant along any row or column.

Minors and cofactors:

Minor of an element a_{ij} is the determinant obtained by deleting ith row and jth column denoted by M_{ij}.

Example.

Properties of determinant:

(a) The value of a determinant remains unchanged by changing rows to columns and columns to rows.

(b) The interchange of two adjacent rows or columns of determinant changes the sign of a determinant without changing its numerical value

(c) If two rows or columns of a determinant are identical then the value of determinant is zero.

(d) If every element of any row or column is multiplied by a factor, then the determinant is multiplied by that factor.

(e) If every element of any row (or column) of a determinant is expressed as sum of two or more numbers, then the determinant can be expressed as the sum of two or more determinants.

(f) A determinant remains unchanged by adding k times the elements of any row (or column) to corresponding elements of any other row (or column) where k is any number or function.

Cramer’s rule to solve a system of linear equations:

Let

a_{1}x + b_{1}y + c_{1}z = d_{1}

a_{2}x + b_{2}y + c_{2}z = d_{2}

a_{3}x + b_{3}y + c_{3}z = d_{3}

is system of linear equations

Note:

(i) If D_{1} = D_{2} = D_{3} = D = 0 then the system has infinitely many solutions.

(ii) If D = 0 and atleast one of D_{1}, D_{2} or D_{3} is not zero then the system has no solution.