Chapter at a Glance
1. Determinant : To every Square matrix A = [aij] of order n associated with anumber (real or complex) called determinant of A. It is denoted by det (A) or |A|.A determinant is also denoted by Del.
2. Properties of Determinant:
(i) The value of the determinant does not change when rows and columns are interchanged. The determinant obtained by interchanging the rows into corresponding columns and vice-versa is called the transpose of the determinant and is denoted by DT. Thus D = DT.
(ii) If all the elements of a row (or column) are zero, then the value of the determinantis zero.
(iii) The interchange of any two rows (or column) of the determinant changes the sign of the determinant. Thus if D* is the new determinant obtained on interchanging any two rows (or columns), then
If i-th and j-th row are interchanged then this operation is denoted by Ri <–> Rj.
(iv) If each element of a row (or column) of a determinant are multiplied by a nonzero constant, then the determinant gets multiplied by the same constant. Thus if we apply Ri ->pRi, i.e, each element of i-th row is multiplied by p to a determinant D, then we get new determinant
(v) If all the elements of a row (or column) are proportional (or identical) to the elements of some other row (or column) then the value of the determinant is zero.
(vi) If each element of a row (or column) of a determinant is zero then, its value is zero.
(vii) If to each element of a row (or a column) of a determinant the equimultiples of corresponding elements of other rows (or columns) are either added or subtracted then value of determinant remains the same.
(viii) If sum of all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants
3. Area of a Triangle : Area of a triangle whose vertices are (x1, y1) , (x2 , y2) and(x3 , y3)
If area of a triangle is given, then use both positive and negative values of the determinant for calculation.
4. Minors and Cofactors :
(i) Minors – The determinant obtained by deleting the i-the row and j-th column passing through the element aij is called the minor of element aij and is denoted by Mij.
(ii) Cofactors – The cofactor of element aij is (–1)i + j times the determinant obtained by deleting the i-th row and jth column passes through aij and is denoted by Aij i.e. Aij = (–1)i + j Mij
5. Value of a Determinant : The sum of the products of each element of any row (or column) by the corresponding co-factors is equal to the value of the determinant.
6. Adjoint of a Matrix : The adjoint of a square matrix A is the transpose of the matrix which is obtained by replacing each element of matrix A by their cofactors. In other words, adjoint of a square matrix A = [aij]n × n is defined as the transpose of the matrix [Aij]n × n, where Aij is the cofactor of the element aij. Adjoint of the matrix A is denoted by adj A
7. If A is any given square matrix of order n, then
where I is the identity matrix of order n.
8. Singular and Non-Singular Matrix
(i) A square matrix A is said to be singular, if |A| = 0
(ii) A square matrix A is said to be non-singular, if |A| ¹ 0
(iii) If A and B are two non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.
9. Invertible Matrix A square matrix
A is invertible if and only if A is non-singular matrix.
10. Inverse of a Square Matrix
If two invertible matrices A and B are such that AB = I or BA = I, where I is the identity matrix then A is called the Inverse matrix of matrix B or B is called the Inverse matrix of matrix A. Inverse of a
11. Application of Determinants and Matrices
We use the determinants and matrices for solving the system of linear equations in two and three variables and checking the consistency of the system of linear equations.
(i) Consistent System : A system of equation is said to be consistent if its solution (one or more) exists.
(ii) Inconsistent System : A system of equation is said to be inconsistent if its solution does not exist.
(iii) Solution of System of Linear Equations using Inverse of a Matrix
Consider the system of linear equations
This matrix equation provides unique solution for the given system of equations as inverse of a matrix is unique. This method of solving system of equations is known as matrix method.
Case-II: If |A| = 0 (i.e. A is singular matrix), then we calculate (adj A) × B If (adj A)B ¹ 0, (0 being zero matrix), then solution does not exist and the system of equation is called consistent. If (adj A)B = 0, then system may be either consistent or inconsistent according as the system have either infinitely many solutions or no solution.
Note: In this chapter, we have to study system of linear equations having unique solutions only
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