A matrix is an arrangement of numbers or symbols in rows and columns. Matrices can be very useful tools for the solution of systems of linear equations.
The number of rows and columns define the size of a matrix. The following matrix, for example, is a 3 × 2 matrix:
$\begin{bmatrix}1 & 2 \\0 & -1\\3 & -2\\ \end{bmatrix}$
The most common way to denote a matrix is by upper-case letters (A, B, C, etc). And the elements of the matrix are denoted by lower-case letters, with two subscript indices for the row and column positions. For example, the element in line "i" and column "j" of a matrix A is denoted as $a_{ij}$.
Addition and Subtraction of Matrices
Only matrices of the same size (same number of rows and columns) can be added or subtracted.
In order to add or subtract two matrices, we have to add or subtract the corresponding elements in both matrices. For example:
$\begin{bmatrix}1 & 2 \\0 & 1\\3 & 2\\ \end{bmatrix}$+$\begin{bmatrix}1 & 1 \\1 & 2\\2 & 1\\ \end{bmatrix}$=$\begin{bmatrix}(1+1) & (2+1) \\(0+1) & (1+2)\\(3+2) & (2+1)\\ \end{bmatrix}$
$\begin{bmatrix}1 & 2 \\3 & 6\\2 & 9\\ \end{bmatrix}$-$\begin{bmatrix}1 & 1 \\1 & 2\\2 & 1\\ \end{bmatrix}$=$\begin{bmatrix}(1-1) & (2-1) \\(3-1) & (6-2)\\(2-2) & (9-1)\\ \end{bmatrix}$
Multiplication of a number by a Matrix
In order to multiply a number by a matrix, we have to multiply the number by all the elements of the matrix. For example:
$3.\begin{bmatrix}2 & -1 \\1 & 2\\3 & 0\\ \end{bmatrix}$=$\begin{bmatrix}6 & -3 \\3 & 6\\9 & 0\\ \end{bmatrix}$
Multiplication of two Matrices
In order to Multiply two matrices the number of columns of the first matrix must be equal to the number of rows of the second matrix.
Consider two matrices A and B, and P=(A)(B). Element $p_{ij}$ is the element in line "i" and column "j" in matrix P. Thus:
$p_{ij}=a_{i1}*b_{1j}+a_{i2}*b_{2j}+...a_{in}*b_{nj}$
That is, element $p_{ij}$ is the sum of the products of all elements in line "i" of the first matrix by all elements in column "j" of the second matrix.
For example:
$\begin{bmatrix}1 & 2 & 3\\4 & 5& 6\\ \end{bmatrix}$.$\begin{bmatrix}1 & 2 \\3 & 4\\ 5 & 6\\\end{bmatrix}$=
$\begin{bmatrix}1.1+2.3+3.5&1.2+2.4+3.6 \\4.1+5.3+6.5&4.2+5.4+6.6\\ \end{bmatrix}$=
$\begin{bmatrix}22&28 \\49&64\\ \end{bmatrix}$
Therefore, the product of two matrices will have the same number of rows of the first matrix and the same number of columns of the second matrix.
Properties of Matrix Multiplication
Consider A, B and C three matrices. The multiplication of matrices has the following properties:
1) $(A)(B)(C)=A(BC)=(AB)C$
2) $(A+B)C=AC+BC$
3) $AB \neq BA$
Identity Matrix
An Identity Matrix ($I$) is a square matrix in which the elements of the main diagonal are equal to 1, and all other elements are equal to 0.
This type of matrix is called Identity, because the product of its multiplication by another matrix $A$ is $A$ itself, that is, $I(A)=A$.
For example:
$\begin{bmatrix}1 & 2\\4 & 5\\ \end{bmatrix}$.$\begin{bmatrix}1 & 0 \\0 & 1\\ \end{bmatrix}$=
$\begin{bmatrix}1.1+2.0 & 1.0+2.1\\4.1+5.0 & 4.0+5.1\\ \end{bmatrix}$=
$\begin{bmatrix}1 & 2\\4 & 5\\ \end{bmatrix}$
Inverse Matrix
An Inverse Matrix ($A^{-1}$) of matrix $A$ is a matrix such that:
$A.A^{-1}=A^{-1}.A=I$, where $I$ is the Identity Matrix.
Transpose Matrix
The transpose of matrix $A$ is the matrix $A^t$, such that, the first column of matrix $A^t$ is equal to the first line of matrix $A$; the second column of matrix $A^t$ is equal to the second line of matrix $A$; and so forth.
For example, consider the following matrix:
$\begin{bmatrix}1 & 2 & 3\\4 & 5& 6\\ \end{bmatrix}$
Its transpose is:
$\begin{bmatrix}1 & 4\\2 & 5\\3& 6\\ \end{bmatrix}$
The Transpose Matrix has the following properties:
P1: $(A^t)^t=A$
P2: $(A+B)^t=A^t+B^t$
P3: $(A.B)^t=B^t.A^t$
P4: $(k.A)^t=k.A^t$, where $k$ ia a real number.
Symmetric and Skew-symmetric Matrices
A matrix A is Symetric when it is equal to its transpose: $A^t=A$
A matrix A is Skew-symmetric when it is equal to the negative of its transpose: $A^t=-A$
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