Animated Solution for Mathematics - Matrices and Determinants: If the system of equations $(\lambda -1)x+(\lambda -4)y+\lambda z=5$ $\lambda x+(\lambda -1)y+(\lambda -4)z=7$ $(\lambda +1)x+(\lambda +2)y-(\lambda +2)z=9$ has infinitely many solutions, then ${\lambda }^2+\lambda$ is equal to

When dealing with a system of three linear equations that possess infinitely many solutions, we are essentially looking at planes that intersect along a common line or coincide. Algebraically, this dependency forces the determinant of the coefficient matrix, $D$, to be exactly zero.

Our starting determinant is defined as:

Avoid the trap of direct expansion, which invites calculation errors. Instead, we simplify the matrix using column operations to create zeros.

Perform the operation $C_3\to C_3+C_2$. The third column transforms, notably simplifying the third row:

Next, we simplify the rows to reduce the complexity of the variables. By applying $R_2\to R_2-R_1$, the $\lambda$ terms in the second row vanish:

Now, expand along the second row using the cofactor expansion method. This yields the following equation:

As we simplify the expression, the higher-order terms consolidate into a manageable quadratic equation:

Factoring this quadratic gives us $(\lambda -3)(2\lambda +1)=0$, resulting in two potential values: $\lambda =3$ and $\lambda =-\frac{1}{2}$.

Evaluating the requested result based on these roots, we find that for $\lambda =3$, the value is $3^2+3=12$. Thus, the final answer is 12.