`A=[(a,1,0),(1,b,d),(1,b,c)],B=[(a,1,1),(0,d,c),(f,g,h)],U=[(f),(g),(h)],V=[(a^2),(0),(0)]` If there is a vector matrix X, such that `AX = U` has infinitely many solutions, then prove that `BX = V` cannot have a unique solution. If `a f d != 0`. Then,prove that `BX = V` has no solution.

A = [{:( a,0,1),(1,c,b),( 1,d,b) :}]B=[{:( a,1,1),(0,d,c),(t,g,h):}]U=[{:(f),(g),(h):}]V=[{:(a^(2)),(0),( 0) :}] if there is vector matrix X . Such that AX= U has infinitely many solutions ,then prove that BX = V cannot have a unique solutions , If afd ne 0 then prove that BX= V has no solutions

Consider a system of linear equation in three variables x,y,z a_1x+b_1y+ c_1z = d_1 , a_2x+ b_2y+c_2z=d_2 , a_3x + b_3y + c_3z=d_3 The systems can be expressed by matrix equation [(a_1,b_1,c_1),(a_2,b_2,c_2),(c_1,c_2,c_3)][(x),(y),(z)]=[(d_1),(d_2),(d_3)] if A is non-singular matrix then the solution of above system can be found by X = A^(-1)B , the solution in this case is unique. if A is a singular matrix i.e. then the system will have no unique solution if no solution (i.e. it is inconsistent) if Where Adj A is the adjoint of the matrix A, which is obtained by taking transpose of the matrix obtained by replacing each element of matrix A with corresponding cofactors. Now consider the following matrix. A=[(a,1,0),(1,b,d),(1,b,c)], B=[(a,1,1),(0,d,c),(f,g,h)], U=[(f),(g),(h)], V=[(a^2),(0),(0)], X=[(x),(y),(z)] If AX=U has infinitely many solutions then the equation BX=U is consistent if

If the system of equations x+ay=0,az+y=0, and ax+z=0 has infinite solutions,then the value of equation has no solution is -3 b.1 c.0 d.3

Consider a system of linear equation in three variables x,y,z a_1x+b_1y+ c_1z = d_1 , a_2x+ b_2y+c_2z=d_2 , a_3x + b_3y + c_3z=d_3 The systems can be expressed by matrix equation [(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)][(x),(y),(z)]=[(d_1),(d_2),(d_3)] if A is non-singular matrix then the solution of above system can be found by X = A^(-1)B , the solution in this case is unique. if A is a singular matrix i.e. then the system will have no solution (i.e. it is inconsistent) if Where Adj A is the adjoint of the matrix A, which is obtained by taking transpose of the matrix obtained by replacing each element of matrix A with corresponding cofactors. Now consider the following matrix. A=[(a,1,0),(1,b,d),(1,b,c)], B=[(a,1,1),(0,d,c),(f,g,h)], U=[(f),(g),(h)], V=[(a^2),(0),(0)], X=[(x),(y),(z)] The system AX=U has infinitely many solutions if :

The pair of equations 2x-5y+4=0 and 2x+y-8=0 has (a) a unique solution (c) infinitely many solutions (b) exactly two solutions (d) no solution

The pair of equations x+2y+5=0 and -3x-6y+1=0 has (a) a unique solution (c) infinitely many solutions(d) no solution (b) exactly two solutions