Physical applications of matrices and determinants Other than notation devices, I don't see any direct application of matrices/determinants in physics. For example, they are just  a different way to write a partial derivative and determinants find if they can be explicitly solved if written down as simultaneous equations. Calculus, for instance, can be directly applied to physical problems, but I don't know of any other application of matrices other than representing equations in a different notation. And in most of the cases like vector products, you just realise that a huge term can just be written down as a determinant, so it is essentially a notational tool. They are used in tensor calculus, but for similar reasons.
Can someone please guide me on more applications with good sources? 
 A: Applications of matrices:


*

*Matrix (aka quantum) Mechanics, obviously

*Mechanics of deformable solids (where matrices describe stresses)

*Statics (most in engineering contexts), where matrices describe stresses.

*Symmetries (where matrices describe rotations/scaling/translations etc..)

*Coordinate transformations, where matrices describe the transformation a coordinate system undergoes.

*Represantation of (Linear) Operators (related to quantum mechanics but not only)
Determinants:


*

*Measure volumes (in transformations etc..)

*Measure volumes in general sense as measure (for example in Path-integral formulation, in many cases the result is expressed as a determinant of a genearally infinite-dimensional matrix)
A: Lie groups are fundamental for talking about anything related to symmetries in physics on a level of some rigor, and every finite-dimensional Lie group is a matrix group. Consequently, the trace as a basic matrix operation shows up anywhere where invariance on the adjoint action of the group is needed, and the matrices are everywhere.
The Slater determinant is what multi-fermion wave-functions are, and this is not a notational trick, since that wave function is actually the n-fold wedge product of basis vectors on some space, which is (up to normalisation) also what the determinant really is.
A: More use of matrices: 
The moment of inertia tensor needed to describe the rotational motion of rigid bodies
The Pauli matrices for spinn-1/2 (but that example is perhaps included in the Lie group example already mentioned).
A: Surprised nobody mentioned optics, so I will. Matrices are used extensively in geometric optics and polarization.
Examples include the ABCD matrix method for representing the effects of optical elements (e.g lenses and mirrors) in ray optics, and the Mueller matrices/Stokes vectors to represent the effects of polarizers and plates on the polarization of light.
A: 
Other than notation devices ...

Matrices are far more than mere notational devices, but even if they were, don't deride notational devices! More compact notation simplifies writing, simplifies reading, simplifies thinking, and because of that, it enables new ways of thinking.
Think of the progress from marks on a stick to representing numbers by devices such as MMXIV to place value notation such as 2014. Even those marks on a stick were extremely important notational devices. Our inborn number sense is limited to $7\pm 2$. Beyond nine, it's just "many". Those marks on a stick were important in letting our human ancestors herd sheep and cattle, knowing when to plant and when to harvest. Notational devices oftentimes are the key that lets mathematics and humanity advance to the next plateau. Zero? That's just a notational device, literally so in the case of the zero in 2014. Abstract symbols like "1", "4", and "8"? Those are nothing more than notational devices for |, ||||, and |||||||||. The symbols "+", "-", etc.: Those are just notational devices, too. We could be verbose and circumlocutious and use words and Roman numerals instead. Writing 2014+342 is just a bunch of notational devices. 
Conceptually, we could do even complex mathematics using nothing but marks on a stick and the right words. Except we couldn't. Notational devices are important. They let us see the forest for the trees. In the case of matrices, I don't see how physicists could have developed quantum mechanics or how NASA could have gotten to the Moon without them. Quantum mechanics uses infinite dimension matrices. Writing that down using marks on a stick would take a long time. The Kalman filter, while not infinite dimension, is hairy in its own right. The matrix notation is what led to the development of to the signal processing concepts that in turn led to the Kalman filter.
