﻿ Matrix calculus
Operations on Matrix
A=

Operations on square matrix
 Determinant |A| Trace |A|
Operations on any size matrix
 Rank |A|

Go to the matrices equations.

 Matrices overview Determinants Transposed matrix Inverse matrix Rank of a matrix
 Matrices overview ▲
The notation of a matrix of size (m n) is defined as     A(m n) = A(row, column)
A convenient shorthand which offers considerable advantage when working with system of linear equations is by using the matrix notation. Consider the set of linear equations of the form: In matrix notation these equations may be represented as: or     AX = C
 The terms of the matrix can be represented as: Distributive law
Left side      A(B + C) = AB + AC

Right side    (A + B)C = AC + BC
A(m n)     B and C (n p)

A and B (m n)     C(n p)
Associative law
Addition    (A + B) + C = A + (B + C)

Multiplication        (AB)C = A(BC)
(m n)

(n n)
Scalar multiplication (kA)B = A(kB) = k(AB) A(m n)      B(n p)
k any number
Commutative law
Addition       A + B = B + A

Multiplication        not commutative
A and B (m n)

Because    A∙B ≠ B∙A
Other algebric laws

(k, v are constants)
 0 + A = A k(A + B)= kA + kB 1 · A = A (k + v)A = kA + vA 0 · A = 0 k(vA) = (kv)A A + ( A) = 0 kA = 0    →    k = 0   or   A = 0 (  1)A =  A
Matrices powers

(c is constant)  Determinants ▲
Determinants - symbol:     det A or |A|
The result of the determinant of a matrix (n ⨯ n) is a real number.
 Size 2 matrix Size 3 matrix General form to evaluate determinant values: In this formula Min is the determinant of the submatrix of A obtained by deleting its ith row and nth column. The determinant Min is called the minor of the element ain and his size is (n-1) ⨯ (n-1).
 Cofactors of matrix Aij It is convenient to consolidate the quantity (-1)i+j and the minor Mij . We define the cofactor Aij of the element aij in determinant A as: Aij = (-1)i+jMij .
 Determinants properties: Example: Avaluate the determinant: det A = 1(1*2-(-1)(-1))-(-2)(2*2-(-1)(-2))+3(2*(-1)-1*(-2)) det A = 1(2-1)+2(4-2)+3(-2+2) = 5
 Transposed matrix     AT ▲
 Transposed matrix AT Interchange of terms across the main diagonal Transposed matrices properties: Example: Find the transposed of the matrix Note: The transposed size of an m ⨯ n matrix is n ⨯ m.
 Inverse matrix     A-1 ▲
 Inverse matrix A-1 = B The matrix A is inversible if there is a matrix B so that: AB = BA = I then the matrix B is the inversed matrix of A. Matrix  I   is the unit matrix.
 Inversed matrices properties: Example: Find the inverse of matrix A 1. Add the unit matrix at the right:  2. Multiply first row by -2 and add it to the second row then multiply first row by -4 and add it to the third row to obtain: 3. Add second and third rows to obtain: 4. Subtract third row from second row: 5. Finally multiply third row by 2 and add it to the first row and multiply third row by -1 to get the unit matrix: And the inverse of A is: Rank of a matrix A ▲
 Rank of a matrix A A square matrix is said to be nonsingular if its determinant is not zero. The rank of an m ⨯ n matrix is the largest integer r for which a nonsingular r ⨯ r submatrix exists. If A and B be n ⨯ n matrices then: rank(A+ B) ≤ rank A + rank B
 Example:Find the rank of matrix A. 1. Multiply first row by 2 and add it to the second row.2. Multiply first row by -3 and add it to the third row.3. Multiply third row by -1 and add it to the first row to get: 4. Add second row to the third row to obtain: And the r ⨯ r non zero matrix is: And the rank of matrix A is 3.