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**

**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**

**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 3×3 Case. To get the first element in our solution matrix ` c`

_{11}

c_{11}= (a_{11}* b_{11}) + (a_{12}* b_{21}) + (a_{13}* b_{31})

Where a_{ij} and b_{ij} are from matrices **A**, **B** respectively.

**A**)

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

**A**

^{T}

**transpose**of a matrix is switching the rows and columns.

For example:

A = |
a b c d e f g h i |
A^{T} = |
a d g b e h c f i |

**A**)

**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 j_{1} j_{2} … j_{n} of the set **S**={1, 2, …, n }.

The sign is + or – according to whether the permutation is even or odd.

**Example:** In our 3×3 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 3×3 matrix would have 6 calculations (3!) to make, whereas a 20×20 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.

**A**)

**adjoint**of

**A**is the transpose of the matrix whose

*i*th, and

*j*th element is the cofactor A

_{ij}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'), whereA'is the matrix obtained from "omitting" the ith and jth rows, of matrixA.

**A**)

**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)