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I. SYSTEM OF LINEAR EQUATIONS
System Of Linear Equation (In Two Variables) :
(i) Consistent Equations :       Definite & unique solution .       [ intersecting lines ]
(ii) Inconsistent Equation :      No solution .                            [ Parallel line ]
(iii) Dependent equation :        Infinite solutions .                    [ Identical lines ]


System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced

 

Cramer's Rule :[ Simultaneous Equations Involving Three Unknowns ]
Let a1x + b1y + c1z = d1 ..... (I) ;

a2x + b2y + c2z = d2 ..... (II) ;

a3x + b3y + c3z = d3 ..... (III)
System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced


Note :
System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced
Remember that if a given system of linear equations have Only Zero Solution for all its variables then
the given equations are said to have Trivial Solution.


Solving System of Linear Equations Using Matrices :
System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced

Then the above system can be expressed in the matrix form as AX = B.
The system is said to be consistent if it has atleast one solution.


(i) System of Linear Equations And Matrix Inverse :
If the above system consist of n equations in n unknowns, then we have AX = B where A is a
square matrix. If A is non–singular, solution is given by X = A–1B.
If A is singular, (adj A) B = 0 and all the columns of A are not proportional, then the system has
infinitely many solutions.
If A is singular and (adj A) B System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced 0, then the system has no solution (we say it is inconsistent).


(ii) Homogeneous System and Matrix Inverse :
If the above system is homogeneous, n equations in n unknowns, then in the matrix form it is
AX = O. ( System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced in this case b1 = b2 =..... bn = 0), where A is a square matrix.
If A is non–singular, the system has only the trivial solution (zero solution) X = 0
If A is singular, then the system has infinitely many solutions (including the trivial solution) and hence it has non–trivial solutions.


(iii) Elementary Row Transformation of Matrix :
The following operations on a matrix are called as elementary row transformations.
(a) Interchanging two rows.
(b) Multiplications of all the elements of row by a nonzero scalar.
(c) Addition of constant multiple of a row to another row.


Note : Similar to above we have elementary column transformations also.


Remark : Two matrices A & B are said to be equivalent if one is obtained from other using elementary
transformations. We write A ~ B.


(iv) Echelon Form of A Matrix : A matrix is said to be in Echelon form if it satisfies the following
(a) The first non-zero element in each row is 1 & all the other elements in the corresponding
column (i.e. the column where 1 appears) are zeroes.
(b) The number of zeros before the first non zero element in any non zero row is less than
the number of such zeroes in succeeding non zero rows.


(v) System of Linear Equations : Let the system be AX = B where A is an m × n matrix, X is
the n–column vector & B is the m–column vector. Let [AB] denote the augmented matrix (i.e.
matrix obtained by accepting elements of B as n + 1th column & first n columns are that of A).



Ex.25 Solve the equations 

System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced


Sol.

System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced


Ex.26 Solve
System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced


Sol.


System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced
And we see that the system has an infinite number of solutions. Specific solutions can be generated by
choosing specific values for k.

 


Ex.27 Number of triplets of a, b & c for which the system of equations ax - by = 2a - b and (c + 1)x + cy = 10 - a + 3 b has infinitely many solutions and x = 1, y = 3 is one of the solutions is


Sol.

System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced

 


Ex.28 Solve System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced


Sol.
System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced
Ex.29 Solve System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced
by reducing the augmented matrix of the system to reduced row echelon form.


Sol.

System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced
System of Linear Equations | Mathematics (Maths) for JEE Main & AdvancedIt is easy to see that x1 = 1, x2 = –3, x3 = 6. The process of solving a system by reducin
the augmented matrix to reduced row echelon form is called Gauss–Jordan elimination.


Ex.30 Determine conditions on a, b and c so that System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced  will have no solutions or have an infinite number of solution.


Sol.
System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced


J. INVERSE OF A MATRIX
(i) Singular & Non Singular Matrix : A square matrix A is said to be singular or non–singular according as |A| is zero or non–zero respectively.


Ex.31 Show that every skew-symmetric matrix of odd order is singular.


Sol.

System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced


(ii) Cofactor Matrix & Adjoint Matrix : Let A = [aij]n be a square matrix. The matrix obtained by
replacing each element of A by corresponding cofactor is called as cofactor matrix of A,
denoted as cofactor A. The transpose of cofactor matrix of A is called as adjoint of A, denoted
System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced

(iii) Properties of Cofactor A and adj A :
System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced
(iv) Inverse of A Matrix (Reciprocal Matrix) : Let A be a non–singular matrix. Then the matrix
System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced

Remarks :
System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced


Characteristic Polynomial & Characteristic Equation : Let A be a square matrix. Then the
polynomial |A – xI| is called as characteristic polynomial of A & the equation |A – xI| = 0 is called as
characteristic equation A.


Remark : Every square matrix A satisfies its characteristic equation (Cayley – Hamilton Theorem).
i.e.

System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced


Ex.32 Find the adjoint of the matrix A =System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced


Sol.

System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced
System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced


Ex.33 If A and B are square matrices of the same order, then adj (AB) = adj B. adj A.


Sol.

System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced

Ex.34 If A be an n-square matrix and B be its adjoint, then show that Det (AB + KIn) = [Det (A) + K]n, where K is a scalar quantity.


Sol.

System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced


Ex.35 If  System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced   be the direction cosines of three mutually perpendicular lines referred to an orthogonal Cartesian co–ordinate system, then prove thatSystem of Linear Equations | Mathematics (Maths) for JEE Main & Advancedis an orthogonal matrix.


Sol.

System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced

Ex.36 Obtain the characteristic equation of the matrix A =System of Linear Equations | Mathematics (Maths) for JEE Main & Advancedand verify that it is satisfied y A and hence find its inverse.


Sol.

System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced

System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced

Ex.37 Find the inverse of the matrix A =  System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced


Sol.

System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced

 

Ex.38 If a non-singular matrix A is symmetric, show that A–1 is also symmetric.


Sol.

System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced

The document System of Linear Equations | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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FAQs on System of Linear Equations - Mathematics (Maths) for JEE Main & Advanced

1. What is a system of linear equations?
Ans. A system of linear equations is a set of two or more equations with the same variables. The solution to the system is the set of values for the variables that satisfy all the equations simultaneously.
2. How can I solve a system of linear equations?
Ans. There are several methods to solve a system of linear equations, such as graphing, substitution, and elimination. Graphing involves plotting the equations on a coordinate plane and finding the point(s) of intersection. Substitution involves solving one equation for one variable and substituting it into the other equation. Elimination involves adding or subtracting the equations to eliminate one variable and solve for the remaining variable.
3. Can a system of linear equations have no solution?
Ans. Yes, a system of linear equations can have no solution. This occurs when the equations are inconsistent and do not intersect. For example, if the lines represented by the equations are parallel, they will never intersect and the system will have no solution.
4. Can a system of linear equations have infinitely many solutions?
Ans. Yes, a system of linear equations can have infinitely many solutions. This occurs when the equations are dependent and represent the same line. In this case, any point on the line will satisfy both equations, resulting in infinitely many solutions.
5. What is the importance of solving systems of linear equations?
Ans. Solving systems of linear equations is important in various fields, such as physics, engineering, economics, and computer science. It allows us to find the intersection points of multiple lines or planes, determine the values of variables in a system, and solve real-world problems involving multiple variables.
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