Question

Explain all basics of determinants class 12 (15 marks)

write in minimum 1 page or more .

Answer 1

Answer:

Determinant: Determinant is the numerical value of the square matrix. So, to every square matrix A = [aij] of order n, we can associate a number (real or complex) called determinant of the square matrix A. It is denoted by det A or |A|.

✨Note

(i) Read |A| as determinant A not absolute value of A.

(ii) Determinant gives numerical value but matrix do not give numerical value.

(iii) A determinant always has an equal number of rows and columns, i.e. only square matrix have determinants.

Value of determinant of a matrix of order 3, A = ⎡⎣⎢a11 a21 a31 a12 a22 a32 a13 a23 a33⎤⎦⎥ is given by expressing it in terms of second order determinant. This is known as expansion of a determinant along a row (or column).

Minor: Minor of an element ay of a determinant, is a determinant obtained by deleting the ith row and jth column in which element ay lies. Minor of an element aij is denoted by Mij.

✨Note: Minor of an element of a determinant of order n(n ≥ 2) is a determinant of order (n – 1).

1).Cofactor: Cofactor of an element aij of a determinant, denoted by Aij or Cij is defined as Aij = (-1)i+j Mij, where Mij is a minor of an element aij.

✨Note

(i) For easier calculations of determinant, we shall expand the determinant along that row or column which contains the maximum number of zeroes.

(ii) While expanding, instead of multiplying by (-1)i+j, we can multiply by +1 or -1 according to as (i + j) is even or odd.Let A be a matrix of order n and let |A| = x. Then, |kA| = kn |A| = kn x, where n = 1, 2, 3,...

Singular and non-singular Matrix: If the value of determinant corresponding to a square matrix is zero, then the matrix is said to be a singular matrix, otherwise it is non-singular matrix, i.e. for a square matrix A, if |A| ≠ 0, then it is said to be a non-singular matrix and of |A| = 0, then it is said to be a singular matrix.

✨Theorems

Theorems(i) If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.

Theorems(i) If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.(ii) The determinant of the product of matrices is equal to the product of their respective determinants, i.e. |AB| = |A||B|, where A and B are a square matrix of the same order.

Theorems(i) If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.(ii) The determinant of the product of matrices is equal to the product of their respective determinants, i.e. |AB| = |A||B|, where A and B are a square matrix of the same order.Adjoint of a Matrix: The adjoint of a square matrix ‘A’ is the transpose of the matrix which obtained by cofactors of each element of a determinant corresponding to that given matrix. It is denoted by adj(A).

Theorems(i) If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.(ii) The determinant of the product of matrices is equal to the product of their respective determinants, i.e. |AB| = |A||B|, where A and B are a square matrix of the same order.Adjoint of a Matrix: The adjoint of a square matrix ‘A’ is the transpose of the matrix which obtained by cofactors of each element of a determinant corresponding to that given matrix. It is denoted by adj(A).In general, adjoint of a matrix A = [aij]n×n is a matrix [Aji]n×n, where Aji is a cofactor of element aji.

Answer 2

Answer:

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