Matrixes

• Matrix is a collection of data that is represented as entries organised in rows and columns.
• A matrix can be described by stating its order which tells you the number of rows and columns. The order is given in $m\times n$ format, where m is a number of rows and n is a number of columns
• Matrix with equal amount of rows and columns is called square matrix
• Matrix with 1 column is called column matrix
• matrix with 1 row is called a row matrix
• If all elements of matrixes are equal and placed at the same places, then matrixes are equal.

  Or, if $a_{ij}=$$b_{ij}$ for all i and j, then matrix A is equal to matrix B
• To add or substract matrixies you need to add or substract the corresponding elements. The order of matricies must be the same
• To multiply a matrix by a scalar you multiply all elements in the matrix by the scalar.
• The elements of the product matrix P=A*B are found by adding the products of elements in row i of matrix A and the elements in column j in matrix B
• The product P = AB of an m*n matrix A and a k*l matrix B exists only if n = k.

  The order of the matrix P is m*l.
• Determinant of a matrix A is:

  $A=$$\begin{bmatrix}a & b \\ c & d\end{bmatrix}=$$detA=$$|A|=$$ad-bc$
• Minor is a determinant, obtained from matrix A by removing row, in which a minor is, and a column, where it is.

  $M_{11}=$$\begin{bmatrix}a_{22} & a_{23} \\ a_{32} & a_{33}\end{bmatrix}$
• $cofactor\, of \,a_{ij} =$$C_{ij} =$$(-1)^{i+j}M_{ij}$
• Determinant of a matrix with 3x3 range is calculated:

  $A=$$\begin{bmatrix}a & b &c\\ d & e & f \\ g&h&i\end{bmatrix}=$$a*$$det\begin{bmatrix}e & f \\ h & i\end{bmatrix} - b*$$det\begin{bmatrix}d & f \\ g & i\end{bmatrix}+$$c*$$det\begin{bmatrix}d & e \\ g & h\end{bmatrix}$
• Identity matrix is a square matrix in which each elements of its principal diagonal is a 1 and each of the other elements is a 0. Examples:

  $\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}, \begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$
• For any matrix A of order n*n, multiplication of A and identity matrix is equal to A:

  $AI=$$IA=$$A$
• When we multiply a matrix by its reciprocal we get the identity matrix (which like '1' for matrices)

  $A *$$A^{-1}=$$I$
• The inverse of $A$is $A^{-1}$ only when:

  $AA^{-1} =$$A^{-1}A=$$I$
• Sometimes there is no inverse matrix.
• For 2x2 matrix inverse matrix is:

  $\begin{bmatrix}a & b \\ c & d\end{bmatrix}^{-1} =$$\frac{1}{ad-bc}\begin{bmatrix}d & -b \\ c & a\end{bmatrix}$
• Transpose of a matrix is simply a flipped version of an origin matrix. Transposition can be done by switching rows and columns of a matrix:

  $A =$$\begin{bmatrix}a & b & c\\ d & e & f\end{bmatrix}, A^T =$$\begin{bmatrix}a & d \\ b & e \\ c & f\end{bmatrix}$
• Rank of matrix is equal to the number of linearly independant rows (or columns) in a matrix. So, it can't have more than matrix's min number of rows or columns.
• The rank of a matrix is usually denoted as $rank(A)\,or\,rk(A)$
• If a matrix has all rows with zero elements, then the rank of a matrix is said to be zero.
• Rank of a matrix can be calculated with 3 methods:

  • Minor method
  • Using Echelon form
  • Using Normal form
• How works minor method of finding rank of a matrix?

  1. We need to calculate determinant. If it is not equal to 0, then rank of a matrix is equal to the order of a matrix
  2. If det(A) = 0, then the rank of the matrix is equal to order of the maximum possible non zero minor of the matrix
  3. If there are not zero elements in a matrix, then matrix rank is bigger than 1 and less then the number of rows or columns (depends from which is smaller)
  4. Than we check all minors of 2nd rank, until we find at least 1 not equal to 0, then next rank minors, until we find rank where all minors are 0.
• Rank of a matrix for rows is equal to the rank of such matrix by columns
• First step of finding inverse matrix process (with the method of algebraic additions): Find the determinant, if it is not equal to 0, then there is an inverse matrix.
• Second step of the process of finding inverse matrix with algebraic additions method: create matrix of minors.
• Third step of the process of finding inverse matrix with algebraic additions method: Create matrix of co-factors and transpose it.
• Fourth step of the process of finding the inverse matrix with the method of algebraic additions: Multiply the transposed matrix to the 1, divided by determinant.
• A matrix is orthogonal when $A^{T}A=$$I, (or\,A^{T}=$$A^{-1})$
• The determinant of orthogonal matrix is either -1 or 1