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Page 4 -- Matrices

A matrix is a convenient notation for the representation of ordered sequences of numbers or variables. Combinations of matrices can represent complex equations in a simple format.

For example, a 3 x 2 matrix represents the union of three lines in 2D space. A 3 x 3 matrix describes the combination of three lines in 3D space.

The properties of matrices are similar to vectors. For example, matrix multiplication is simply the sum of the product of the corresponding row in A and column in B. The size of the resulting matrix is determined by the number of columns in A and rows in B.

Equation

In addition, matrices have a unique set of properties.

  • The identity matrix, [I], is defined such that each element on the main diagonal equals one. Therefore, multiplying by the identity matrix does not change the initial matrix.
    Equation

  • Another special matrix property is the transpose. A matrix can be transposed by exchanging all rows and columns.
    Equation

  • A matrix [A] is invertible if there exists a matrix [B] such that [B][A]=[I] and [A][B]=[I]. If [B] exists, then it is called the inverse of [A]. The inverse is useful for matrix division, as follows:

[A][B]=[C]

[A]-1[B]=[A]-1[C]

[A]-1[A]=[I]


[I][B]=[A]-1[C]

[B]=[A]-1[C]