# What is the role of determinant and trace of matrices in physics? [closed]

There is vast area of physics where we have to use matrices.It is not only to do the mathematical problems in physics but also to produce a physical realization of an operation. I think matrices carry a huge amount of physics in symmetry operations. Again a matrix can be described by two numbers one is determinant and another one is trace. My question is what are the physical significances of DETERMINANT as well as TRACE?

## closed as too broad by Emilio Pisanty, stafusa, ACuriousMind♦Oct 23 '17 at 23:42

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• Well, this is different on a case by case scenario. Matrices are indeed representations of operators (that are widely used in, say, Quantum Mechanics) but still this will be quite a narrow description of the why. – gented Oct 23 '17 at 15:23
• A matrix certainly cannot be described only by its determinant and its trace. For instance $$P_{12}=\left(\begin{array}{ccc} 0&1&0\\ 1&0&0\\ 0&0&1\end{array}\right)\, \qquad P_{13}= \left(\begin{array}{ccc} 0&0&1\\ 0&1&0\\ 1&0&0\end{array}\right)$$ have the same determinant and trace and are obviously distinct. – ZeroTheHero Oct 23 '17 at 15:24
• My question is not about the use of matrices in Quantum Mechanics or somewhere. – DIPANJAN HAZRA Oct 23 '17 at 15:33
• What is your question then? Your title is " Why the use of matrices is so important in physics". Moreover, you claim "Again a matrix can be described by two numbers one is determinant and another one is trace" which is incorrect. – ZeroTheHero Oct 23 '17 at 15:36
• The trace and the determinant are but two of a number of quantities invariant under conjugation of a matrix by a unitary transformation, i.e. under a change of basis. This is hardly enough to completely specify a matrix. For instance in $3\times 3$ there is another invariant (see math.stackexchange.com/a/807183/160660) . – ZeroTheHero Oct 23 '17 at 19:28

Well, there is not much to tell. The more physical you can get with determinant is the following: the determinant represents the "volume distorsion", which means it only tells you by how much your linear transformation will change the volume of a parallelogram. For instance, the matrix $2\mathbb{1}_{2\times2}$ tells you that the square of area 1 will be stretched to a square of area 4 (because $\mathrm{det}(2\mathbb{1}_{2\times2})=4$). More generally, this is true for any dimension, and also for every kind of linear transformation. Think of the jacobian when performing a coordinate transformation in an integral. The measure has to change since the coordinate transformation may change the volume of an infinitesimal n-dimensional parallelogram. Also, that is why we use in quantum mechanics SO(n) or SU(n) as symmetry groups; their determinant is 1 so that the volume is "conserved" when we rotate things.
I can think about $2\times 2$ matrices to describe a set of two linear equations. These systems of two linear equations are extremely common in physics.
Here is a useful relation between determinant and trace. The linear Lie Group transformations done in physics are of the form $M=e^\Theta$ where both $M$ and $\Theta$ are matrices. Then it is true that: $$det(M)=1 \quad if \ and \ only \ if \quad Trace(\Theta)=0$$ This is how the det=1 condition, signified by the "S" in SO(n), SU(n), and SL(n), turns into a constraint on the coordinates and generators of the group.