we have $ \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & \pi \end{bmatrix} $$ \begin{bmatrix} x \ y \ z \end{bmatrix} $ = $ \begin{bmatrix} 6 \ 10 \ \mu \end{bmatrix} $ By $R_2- R_1, R_3- R_2$ $ \begin{bmatrix} 1 & 1 & 1 \ 1 & 2 & 3 \ 1 & 2 & \pi-3 \end{bmatrix} $$ \begin{bmatrix} x \ y \ z \end{bmatrix} $ = $ \begin{bmatrix} 6 \ 10 \ \mu-10 \end{bmatrix} $ (i) The system has unique solution if the coefficient matrix is non-singular (or the rank A, r= the number of unknowns , n =3). This requires λ-3 not equal to 0, Hence λ is not equal to 3. Hence the system has unique solution. (ii) If λ=3 the coefficient matrix and the augmented matrix becomes $ \begin{bmatrix} 1 & 1 & 1 \ 0 & 1 & 2 \ 0 & 0 & 0 \end{bmatrix} $ and $ \begin{bmatrix} 1 & 1 & 1 & 6 \ 0 & 1 & 2 & 4 \ 0 & 0 & 0 & \mu-10 \end{bmatrix} $

The rank of A = 2 the rank of [A,B] will be also 2 if μ = 10.

Thus if λ=33and μ = 10 thesystem is consistent. But the rank of A (= 2) is less than the number of unknowns (=3). Hence the equation will posses infinite solutions.

(iii) If λ=3 and μ≠10 , the rank of A=2, and the rank of [A,B] = 3. They are not equal and the equations will be inconsistent and will not posses any solution.