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:
Linear set of equationsn
In matrix notation these equations may be represented as: Linear set of equationsn in matrix form or     AX = C
The terms of the matrix can be represented as: Short representation of matrices
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)
Matrices powers Matrices powers
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.