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Determinants Class 12 Notes Mathematics
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CBSE Class 12 Mathematics Chapter 4 Determinants
- A determinant of a square matrix A is denoted by det.A or |A|.
- A determinant of order 1 x 1 matrix is given by
- A determinant of order of 2 x 2 matrix A is given by
- A determinant of order 3 x 3 matrix A is given by (expanding along ( )
- We can find the value ofa a determinant by expanding along any one of the three rows (or columns) and the value remains same.
- Generally, we find the value of a determinant by expanding along a row or column which has maximum number of zeroes.
- For any square matrix A, the |A| satisfy following properties.
- |A′| = |A|, where A′ = transpose of A.
- 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 equimultiples of corresponding elements of other rows or columns are added, then value of determinant remains same.
- If A is skew symmetric matrix of odd order, then |A| = 0.
- Area of a triangle with vertices is given by
- Minors: Minor of an element aij of the determinant of matrix A is the determinant obtained by deleting row and column and denoted by
- Confactors: Cofactor of aij of given by Aij = (– 1)i+ j Mij.
- Value of determinant of a matrix A is obtained by sum of product of elements of a row (or a column) with corresponding cofactors. For example, .
- If elements of one row (or column) are multiplied with cofactors of elements of any other row (or column), then their sum is zero. For example,
- Adjoint of a matrix: where A is square matrix of order n.
- Singular Matrix: A square matrix A is said to be singular or non-singular according as
- Inverse of a square matrix: If AB = BA = I, where B is square matrix, then B is called inverse of A. Also
- A square matrix A has inverse if and only if A is non-singular.
then these equations can be written as AX = B, where
- Unique solution of equation AX = B is given by
- 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
• | there exists unique solution
• |A| = 0 and (adj A) B ≠ 0, then there exists no solution
• then system may or may not be consistent and has infinite solutions.
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