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 :

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

Solve the following system of equations by using determinants: x+y+z=1 , a x+b y+c z=k , a^2x+b^2y+c^2z=k^2 .

Solve the following system of equations by using determinants: x+y+z=1 , a x+b y+c z=k , a^2x+b^2y+c^2z=k^2

Consider the system of equations a_(1) x + b_(1) y + c_(1) z = 0 a_(2) x + b_(2) y + c_(2) z = 0 a_(3) x + b_(3) y + c_(3) z = 0 If |(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| =0 , then the system has

Solve the following system of linear equations by Cramers rule: x+y+z+1=0,\ \ a x+b y+c z+d=0,\ \ a^2x+b^2y+c^2z+d^2=0

Solve the system of the equations: a x+b y+c z=d , a^2x+b^2y+c^2z=d^2 , a^3x+b^3y+c^3z=d^3 .

If a_1x+b_1y+c_1z=0, a_2x+b_2y+c_2z=0, a_3x+b_3y+c_3z=0 and |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|=0 , then the given system has, 1) more than two solutions 2) one trivial and one non trivial solutions 3) no solution 4) only trivial solution (0, 0, 0)

If a_1x^3 + b_1x² + c_1x + d_1 = 0 and a_2x^3 + b_2x^2+ c_2x + d_2 = 0 have a pair of repeated common roots, then prove that |[3a_1,2b_1,c_1],[3a_2,2b_2,c_2],[a_2b_1-a_1b_2,c_1a_2-c_2a_1,d_1a_2-d_2a_1]|=0

|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|=0 then the system of equations a_1x+b_1y+c_1z=0, a_2x+b_2y+c_2z=0, a_3x+b_3y+c_3z=0 has

Solve the following system of linear equations by matrix method: 2x+3y+3z=1,2x+2y+3z=2,x-2y+2z=3