Matrix algebra

Questions
1

I am trying to do matrix rearragement in mathematica, with
Assumptions = DLL [Element] Matrices[{3, 3}];
$Assumptions = DLR [Element] Matrices[{3, 3}];
$Assumptions = DLT [Element] Matrices[{3, 3}];
$Assumptions = DLB [Element] Matrices[{3, 3}];
$Assumptions = DRL [Element] Matrices[{3, 3}];
$Assumptions = DRR [Element] Matrices[{3, 3}];
$Assumptions = DRT [Element] Matrices[{3, 3}];
$Assumptions = DRB [Element] Matrices[{3, 3}];
$Assumptions = DTL [Element] Matrices[{3, 3}];
$Assumptions = DTR [Element] Matrices[{3, 3}];
$Assumptions = DTT [Element] Matrices[{3, 3}];
$Assumptions = DTB [Element] Matrices[{3, 3}];
$Assumptions = DBL [Element] Matrices[{3, 3}];
$Assumptions = DBR [Element] Matrices[{3, 3}];
$Assumptions = DBT [Element] Matrices[{3, 3}];
$Assumptions = DBB [Element] Matrices[{3, 3}];
$Assumptions = QL [Element] Matrices[{3, 1}];
$Assumptions = QR [Element] Matrices[{3, 1}];
$Assumptions = QT [Element] Matrices[{3, 1}];
$Assumptions = QB [Element] Matrices[{3, 1}];

eq[1] = DLL QL + DLR QR + DLT QT + DLB QB;
eq[2] = DRL QL + DRR QR + DRT QT + DRB QB;
eq[3] = DTL QL + DTR QR + DTT QT + DTB QB;
eq[4] = DBL QL + DBR QR + DBT QT + DBB QB;
QR = [Lambda]1 QL; QB = [Lambda]2 QT;
QR = [Lambda]1 QL; QB = [Lambda]2 QT;
FT = DTL QL + DTR QR + DTT QT + DTB QB
eq[6] = DBL QL + DBR QR + DBT QT + DBB QB + [Lambda]2 FT
eq[7] = Collect[eq[6], QL]
QT = E1 (DBL + DBR [Lambda]1 + DTL [Lambda]2 +
DTR [Lambda]1 [Lambda]2) QL (-Inverse[(DBT +DBB [Lambda]2+DTT [Lambda]2+DTB [Lambda]2^2)])
eq[8] = DRL QL + DRR QR + DRT QT + DRB QB + [Lambda]1 FL
[Lambda]2 = 1; QL = 1
eq[9] = Collect[eq[8], [Lambda]1]

The answer I am getting is
DRL + DBL DRB E1 + DBL DRT E1 + DRB DTL E1 +
DRT DTL E1 + (DLL + DRR + DBL DLB E1 + DBL DLT E1 + DBR DRB E1 +
DBR DRT E1 + DLB DTL E1 + DLT DTL E1 + DRB DTR E1 +
DRT DTR E1) [Lambda]1 + (DLR + DBR DLB E1 + DBR DLT E1 +
DLB DTR E1 + DLT DTR E1) [Lambda]1^2

which forms a quadratic eigenvalue problem in \lambda_1 but the order of multiplication of matrices are not coming correct. Can this be fixed

Quadraticeigenvlaueproblem.nb (12 KB)

2

You appear to be confusing Dot (for matrix multiplication and related) with Times, and that latter is commutative (hence can reorder). Also redefining $Assumptions will not be of much use.

3

Thank you for the response
I tried using Dot in QT and QB since later on everything comes as X QL=0 and we get co-eficients from X..
QR = [Lambda]1 QL ; QB = QT;

FL = DLL QL + DLR QR + DLT.QT + DLB .QB;
eq[5] = DRL QL + DRR QR + DRT.QT + DRB .QB + [Lambda]1 FL;

[Lambda]2 = 1
FT = DTL QL + DTR QR + DTT.QT + DTB.QB
eq[6] = DBL QL + DBR QR + DBT.QT + DBB.QB + [Lambda]2 FT
eq[7] = Collect[eq[6], QL]
QT = E1.(DBL + DBR [Lambda]1 + DTL [Lambda]2 +
DTR [Lambda]1 [Lambda]2)
eq[8] = DRL QL + DRR QR + DRT.QT + DRB .QB + [Lambda]1 FL
QL = 1 (QL here i treated as some number since the the equation is X QL=0, where X=eq8]
eq[10] = Collect[Expand[eq[8]], [Lambda]1]

i am getting the result as
DRL + DLR [Lambda]1^2 + [Lambda]1 (DLL + DRR +
DLB.E1.(DBL + DTL + DBR [Lambda]1 + DTR [Lambda]1) +
DLT.E1.(DBL + DTL + DBR [Lambda]1 + DTR [Lambda]1)) +
DRB.E1.(DBL + DTL + DBR [Lambda]1 + DTR [Lambda]1) +
DRT.E1.(DBL + DTL + DBR [Lambda]1 + DTR [Lambda]1).

This result is not expandable. Expand command gives me the same result. So how to get the co-ffiecient after that