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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

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.



For example:

Aa b c d e f g h iAT:a d g b e h c f i


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.


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)