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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.

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.

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

- 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]