I'd like to solve this equation for $A = B$ where $B = I$, which represents 3 systems of 3 linear equations, for $a, b, c, d, e, f$, without writing LinearSolve 3 times. What is a simple way to accomplish this?


   A = {
   {a + b, -a + b, a + 2 b}, 
   {c + d, -c + d, c + 2 d}, 
   {e + f, -e + f, e + 2 b}
B = IdentityMatrix[3]
(*want to solve A == B, but probably wrong...*)
M = LinearSolve[A, IdentityMatrix[3]]
  • $\begingroup$ So you want to invert A ? $\endgroup$ Commented Sep 8, 2012 at 20:38
  • $\begingroup$ @b.gatessucks If A == I then also A^-1 == I. Thus he just wants to find a,b,c,d,e,frather than A^-1. $\endgroup$
    – Artes
    Commented Sep 9, 2012 at 0:44
  • $\begingroup$ @T.Webster Could you explain what you'd expected from an answer, if existing ones were not satisfactory ? $\endgroup$
    – Artes
    Commented Sep 15, 2012 at 23:48
  • $\begingroup$ @Artes Sorry I have been away and thanks for your answer. $\endgroup$
    – T. Webster
    Commented Sep 17, 2012 at 8:08
  • $\begingroup$ The question related to finding the left inverse of some matrix $A$. But a solution I found from my linear algebra course is to use Gaussian elim. (RowReduce) with the augmented matrix $[A I]$. $\endgroup$
    – T. Webster
    Commented Sep 17, 2012 at 8:13

2 Answers 2


As the documentation says : LinearSolve[m,b] finds an x which solves the matrix equation m.x == b, i.e. in your case it finds x such that A.x == B (Dot[A,x] == B). However your task is to find A solving an adequate system of 9 linear equations for 6 variables a,b,c,d,e,f knowing that B is an IdentityMatrix. You are trying to solve an overdetermined system of linear equations and there could exist any solutions only if certain compatibility conditions were satisfied.

For your task use simply Solve :

Solve[A == B, {a, b, c, d, e, f}]


Reduce[A == B, {a, b, c, d, e, f}]

This means that there are no solutions, i.e. the above equation is contradictory. You could use Variables[A] instead of specifying variables {a, b, c, d, e, f}.

Consider a different matrix equation where we have 4 unknowns and 4 independent equations e.g. :

A1 = {{a + b, a - 2 b}, {a - c, c + d}};
Solve[ A1 == IdentityMatrix[2], {a, b, c, d}] 
{{a -> 2/3, b -> 1/3, c -> 2/3, d -> 1/3}}

i.e. there is only one solution.


Inverse[A] could be a solution assuming that you wanted x in the matrix equation A.x == IdentityMatrix[3] when A was given. There exists an inverse matrix to A under this condition :

Det[ A] != 0


-4 b^2 c + 4 a b d + 4 b c f - 4 a d f != 0

Neither A exists nor this assumption can be satisfied when A is defined as in your question and B is an identity matrix.


Use Inverse to invert a matrix :

bigA = {{a + b, -a + b, a + 2 b}, {c + d, -c + d, c + 2 d}, {e + f, -e + f, e + 2 b}};


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.