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LINEAR ALGEBRA OR MATRIX MATHEMATICS

Matrix definitions are provided in a separate section, called Matrices. A matrix is referenced by its elements:

A=⎡⎢⎢⎢⎣ a₁₁ a₁₂ a₁₃ a₂₁ a₂₂ a₂₃ a₃₁ a₃₂ a₃₃ ⎤⎥⎥⎥⎦

This is a 3 × 3 matrix because it is composed of three columns and three rows respectively.The first number represents the number of rows, while the second number represents the number of columns. There can be an infinite number of different arrangements when dealing with matrices, this is the reason why its important to understand the basics before continuing to more complex matrix math.

ADDITION AND SUBTRACTION

Addition and subtraction of a matrix is quite simple. As a rule of subtraction and addition, when attempting to add or subtract matrices it is very important that the numbers of columns and rows of both matrices are equal. In other words, it is impossible to add or subtract a 2 × 2 matrix with a 3 × 3 matrix. For these operations, the operation is performed on each individual element. A₁₂ is added to B₁₂, A₂₂ is added to B₂₂, and so on. Consider matrices A and B where:

A = ⎡⎢⎢⎢⎣ a₁₁ a₁₂ a₂₁ a₂₂ a₃₁ a₃₂ ⎤⎥⎥⎥⎦ B=⎡⎢⎢⎢⎣ b₁₁ b₁₂ b₂₁ b₂₂ b₃₁ b₃₂ ⎤⎥⎥⎥⎦

Adding the matrices yields

A + B = ⎡⎢⎢⎢⎣ a₁₁ + b₁₁ a₁₂ + b₁₂ a₂₁ + b₂₁ a₂₂ + b₂₂ a₃₁ + b₃₁ a₃₂ + b₃₂ ⎤⎥⎥⎥⎦

and similarly B can be subtracted from A,

A - B = ⎡⎢⎢⎢⎣ a₁₁ − b₁₁ a₁₂ − b₁₂ a₂₁ − b₂₁ a₂₂ − b₂₂ a₃₁ − b₃₁ a₃₂ − b₃₂ ⎤⎥⎥⎥⎦

MULTIPLICATION

There are two different types of matrix multiplication, scalar and non scalar. Scalar multiplication refers to the instance where a scalar is multiplying a matrix. Before continuing with this section, one needs to understand that A × B ≠ B × A. Take for example 3 × A. This simply means that every row and column within the matrix will be multiplied by the scalar in front of it as shown below:

3 × A = ⎡⎢⎢⎢⎣ 3a₁₁ 3a₁₂ 3a₁₃ 3a₂₁ 3a₂₂ 3a₂₃ 3a₃₁ 3a₃₂ 3a₃₃ ⎤⎥⎥⎥⎦

When multiplying two matrices together it is important that the inner number of the two matrices are identical. For example, multiplying a 3x3 times a 3x200 matrix will work because the number of columns of the first matrix matches the number of rows of the second matrix. One cannot multiply a matrix with n number of columns with a matrix with m number of rows. Consider the matrices D and C where:

D=⎡⎢⎣ d₁₁ d₁₂ d₂₁ d₂₂ ⎤⎥⎦ C=⎡⎢⎣ c₁₁ c₁₂ c₁₃ c₂₁ c₂₂ c₂₃ ⎤⎥⎦

Thus, D × C = ⎡⎢⎣ d₁₁ × c₁₁ + d₁₂ × c₂₁ d₁₁ × c₁₂ + d₁₂ × c₂₂ d₁₁ × c₁₃ + d₁₂ × c₂₃ d₂₁ × c₁₁ + d₂₂ × c₂₁ d₂₁ × c₁₂ + d₂₂ × c₂₂ d₂₁ × c₁₃ + d₂₂ × c₂₃ ⎤⎥⎦

IDENTITY MATRIX

The identity matrix I_n receives its name because any multiplication with it leaves the matrix unchanged. In order to understand this, consider the following general example with matrix Z.

Z = ⎡⎢⎣ a b c d ⎤⎥⎦

Z × I_n = ⎡⎢⎣ a b c d ⎤⎥⎦ × ⎡⎢⎣ 1 0 0 1 ⎤⎥⎦ = ⎡⎢⎣ a × 1 + b × 0 a × 0 + b × 1 c × 1 + d × 0 c × 0 + d × 1 ⎤⎥⎦ = ⎡⎢⎣ a b c d ⎤⎥⎦

The I_n must be a square, diagonal matrix with elements, 1, and its n subscript is defined by the number of rows or columns it is composed of.