Solve the following system of linear equations in terms of parameter $a\in\mathbb R$ and explain geometric interpretation of this system: $ax+y+z=1,2x+2ay+2z=3, x+y+az=1$.

By Cronecker Capelli's theorem, we get: $$ \begin{bmatrix} a & 1 & 1 & 1\\ 2 & 2a & 2 & 3\\ 1 & 1 & a & 1\\ \end{bmatrix}$$

Row echelon form of this matrix is $$\begin{bmatrix} 1 & 1 & a & 1\\ 0 & 2(a-1) & 2(1-a) & 1\\ 0 & 0 & (1-a)(a+2) & (3-2a)/2\\ \end{bmatrix}$$

System is inconsistent for $a=1 \lor a=-2 $. For every other value of $a$, system has unique solution.

For every $a$ instead of $a=1 \land a=-2$ there are three planes that intersect at a point.

Question: Is this geometric interpretation correct?

  • $\begingroup$ You should also study what relation exists between the planes when $a=1$ and when $a=-2$. Are they parallel? Are two of them parallel and does one of them intersect with these two?... $\endgroup$
    – MasB
    Jul 16, 2016 at 17:28

1 Answer 1


When $a\neq1$ and $a\neq-2$ the solution is $$x=z=\frac{2a-3}{2(a-1)(a+2)},\qquad y=\frac{3a-1}{2(a-1)(a+2)}$$

When $a=1$ the first and third equations are the same and inconsistent with the second one, i.e. the plane described by the first and third equations is parallel to the plane defined by the second equation. The augmented matrix has rank 2 while the coefficient matrix has rank 1.

When $a=-2$ the augmented matrix has rank $3$ and the coefficient matrix has rank 2, so again the three equations are inconsistent. You might find this page useful to work out the geometric interpretation: http://www.vitutor.com/geometry/space/three_planes.html

  • $\begingroup$ To summarize, in case where $a\neq 1 \land a\neq -2$ we have two coincident planes and one that intersects them in a line. In case where $a=1 \land a=-2$ we have two coincident planes and the other is parallel to them. Is this correct? $\endgroup$
    – user300046
    Jul 16, 2016 at 18:41
  • $\begingroup$ No, when $a\neq 1$ and $a\neq 2$, both the augmented matrix and coefficient matrix have rank 3 and so the planes intersect at a point (the one given in my answer). When $a=1$ the two of the planes coincide and are parallel the the third plane. When $a=-2$ each plane cuts the other two in a line. Look at the link in my answer. $\endgroup$
    – smcc
    Jul 16, 2016 at 21:23

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