Linear Algebra Math
Here are some of the definitions and examples used in Linear Algebra and specifically the linear algebra calculators available.
Let A, B, and C represent n x n matrices.
Example of a 3 x 3 Matrix:
1 2 3
1 1 2
0 1 2

A + B = C

The addition of two matrices is straight forward. You just add each matrix positionwise. So the upperleft element of matrix A plus the upperleft element of matrix B is the upperleft element in matrix C. Do the same for all elements.

A x B = C

The multiplication of two matrices is not quite as simple. First we need the matrices to be of proper size. This means matrix A size n x m must be multiplied by a m x p matrix. The resultant matrix will then be n x p. For our case, we are using n x n matrices, so this isn't a problem.
The equation for multiplying two matrices is : (elementwise)
[AB]_{ij} = SIGMA [A]_{ik}[B]_{kj}
Where the SIGMA summation goes from k=1...n
A example element from our 3x3 Case. To get the first element in our solution matrix c11
c_{11} = (a_{11} _ b_{11}) + (a_{12} * b_{21}) + (a_{13} * b_{31})
Where aij and bij are from matrices A, B respectively.

trace(A)

The trace of a matrix is simply the summation of its main diagonal.

AT

The transpose of a matrix is switching the rows and columns.
For example:
A  a b c d e f g h i  AT:  a d g b e h c f i 

det(A)

The determinant of a matrix is not quite simple. For a n x n matrix the definition of the determinant is as follows :
det(A) = SIGMA (±)a_{1j1} a_{2j2}. . .a_{njn}
Where SIGMA is our summation over all permutations j1 j2 ... jn of the set S={1, 2, ..., n }.
The sign is + or  according to whether the permutation is even or odd.
Example: In our 3x3 case it is a little easier, and boils down to :
det(A) = aei + cdh + bfg  ceg  bdi  afh
Where are matrix first row is a b c , 2nd row d e f, and 3rd row, g h i
Calculation Technique: For the n x n the calculation of the determinant, by definition, is based upon a factorial number of calculations with respect to the size of the matrix. ie. a 3x3 matrix would have 6 calculations (3!) to make, whereas a 20x20 matrix would have 2.43 x 10^18 calculations (20!).
So instead of brute forcing the calculations, I first do some operations on the matrix, which converts it to a upper triangular matrix, and then calculate the determinant by multiplying down the diagonal, since everything below is 0, this will give the determinant.

adj(A)

The adjoint of A is the transpose of the matrix whose _i_th, and _j_th element is the cofactor Aij of the _a_ij element from matrix A.
The cofactor of an element _a_ij from matrix A is :
a_{ij} = (1)^{i + j} * det (A'), where A' is the matrix obtained from "omitting" the ith and jth rows, of matrix A.

inv(A)

The inverse of A is the matrix which when multiplied to A returns the identity matrix.
Calculation Technique: The inverse was obtained using the Theorem:
Aadj(A) = det(A)I_{n}
Which when manipulated gives you:
A^{1} = (1 / det(A)) * adj(A)