If the matrix equation x bmatrix 1 2 1 bmatrix + y bmatrix 1 1 -1 bmatrix + z bmatrix 2 -1 bmatrix = bmatrix 1 4 bmatrix has infinitely many solutions for the variables x, y, z , where , R , then the value of ^2 + 2 ^2 + + is equal to :

Step-by-step solution

The given matrix equation can be written as a system of linear equations: x + y + 2z = 1 2x + y - z = 4 x - y + z = For the system to have infinitely many solutions, we must have D = D₁ = D₂ = D₃ = 0 . D = vmatrix 1 & 1 & 2 2 & 1 & -1 1 & -1 & vmatrix = 1( - 1) - 1(2 + 1) + 2(-2 - 1) = - - 8 Setting D = 0 , we get = -8 . Now, D₃ = vmatrix 1 & 1 & 1 2 & 1 & 4 1 & -1 & vmatrix = 1( + 4) - 1(2 - 4) + 1(-2 - 1) = - + 5 Setting D₃ = 0 , we get = 5 . We can verify that for = -8 and = 5 , D₁ = 0 and D₂ = 0 as well. Theref