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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 ]

**Cramer's Rule :[ Simultaneous Equations Involving Three Unknowns ]**

Let a1x + b1y + c1z = d1 ..... (I) ;

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

a3x + b3y + c3z = d3 ..... (III)**Note :****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 :**

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 nonâ€“singular, solution is given by X = A^{â€“1}B.

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 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. ( in this case b_{1} = b_{2} =..... b_{n} = 0), where A is a square matrix.

If A is nonâ€“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 nonâ€“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 Ã— n matrix, X is

the nâ€“column vector & B is the mâ€“column vector. Let [AB] denote the augmented matrix (i.e.

matrix obtained by accepting elements of B as n + 1^{th} column & first n columns are that of A).

**Ex.25 Solve the equations **

**Sol.**

**Ex.26 Solve**

**Sol.**

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.**

**Ex.28 Solve **

**Sol.****Ex.29 Solve ****by reducing the augmented matrix of the system to reduced row echelon form.**

**Sol.**

It is easy to see that x1 = 1, x2 = â€“3, x3 = 6. The process of solving a system by reducin

the augmented matrix to reduced row echelon form is called Gaussâ€“Jordan elimination.

**Ex.30 Determine conditions on a, b and c so that **** will have no solutions or have an infinite number of solution.**

**Sol.**

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

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

**Sol.**

**(ii) Cofactor Matrix & Adjoint Matrix :** Let A = [a_{ij}]_{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

**(iii) Properties of Cofactor A and adj A :****(iv) Inverse of A Matrix (Reciprocal Matrix) :** Let A be a nonâ€“singular matrix. Then the matrix**Remarks :**

**Characteristic Polynomial & Characteristic Equation :** Let A be a square matrix. Then the

polynomial |A â€“ xI| is called as characteristic polynomial of A & the equation |A â€“ xI| = 0 is called as

characteristic equation A.

**Remark :** Every square matrix A satisfies its characteristic equation (Cayley â€“ Hamilton Theorem).

i.e.

**Ex.32 Find the adjoint of the matrix A =**

**Sol.**

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

**Sol.**

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

**Sol.**

**Ex.35 If ** ** be the direction cosines of three mutually perpendicular lines referred to an orthogonal Cartesian coâ€“ordinate system, then prove that****is an orthogonal matrix.**

**Sol.**

**Ex.36 Obtain the characteristic equation of the matrix A =****and verify that it is satisfied y A and hence find its inverse.**

**Sol.**

**Ex.37 Find the inverse of the matrix A =**

**Sol.**

**Ex.38 If a non-singular matrix A is symmetric, show that A ^{â€“1} is also symmetric.**

**Sol.**

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