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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 position-wise. So the upper-left element of matrix A plus the upper-left element of matrix B is the upper-left 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

c11 = (a11 _ b11) + (a12 * b21) + (a13 * b31)

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:

Aa b c d e f g h iAT: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 (±)a1j1 a2j2. . .anjn

    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 :

aij = (-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)In

Which when manipulated gives you:

A-1 = (1 / det(A)) * adj(A)