System of Linear Equations

Trace of a matrix :
Let A=[a_ij] _nxn is a square matrix of order n, then the sum of diagonal elements is called the trace of a matrix which is denoted by tr(A). tr(A) = a₁₁ + a₂₂ + a₃₃+ ……….+ a_nn

Properties of trace of matrix:
Let A and B be any two square matrices of order n, then 

1. tr(kA) = k tr(A) where k is a scalar.
2. tr(A+B) = tr(A)+tr(B)
3. tr(A-B) = tr(A)-tr(B)
4. tr(AB) = tr(BA)

Solution of a system of linear equations:
Linear equations can have three kind of possible solutions: 

• No Solution
• Unique Solution
• Infinite Solution

Rank of a matrix: Rank of matrix is the number of non-zero rows in the row reduced form or the maximum number of independent rows or the maximum number of independent columns. 
Let A be any m x n matrix and it has square sub-matrices of different orders. A matrix is said to be of rank r, if it satisfies the following properties: 

1. It has at least one square sub-matrices of order r who has non-zero determinant.
2. All the determinants of square sub-matrices of order (r+1) or higher than r are zero.

Rank is denoted as P(A).
if A is a non-singular matrix of order n, then rank of A = n i.e. P(A) = n. 

Properties of rank of a matrix:

1. If A is a null matrix then P(A) = 0 i.e. Rank of null matrix is zero.
2. If I_n is the nxn unit matrix then P(A) = n.
3. Rank of a matrix A mxn , P(A) ≤ min(m,n). Thus P(A) ≤m and P(A) ≤ n.
4. P(A _nxn ) = n if |A| ≠ 0
5. If P(A) = m and P(B)=n then P(AB) ≤ min(m,n).
6. If A and B are square matrices of order n then P(AB) = P(A) + P(B) – n.
7. If A_m×1 is a non zero column matrix and B_1×n is a non zero row matrix then P(AB) = 1.
8. The rank of a skew symmetric matrix cannot be equal to one.

System of homogeneous linear equations AX = 0. 

1. X = 0. is always a solution; means all the unknowns has same value as zero. (This is also called trivial solution)
2. If P(A) = number of unknowns, unique solution.
3. If P(A) < number of unknowns, infinite number of solutions.

System of non-homogeneous linear equations AX = B. 

1. If P[A:B] ≠P(A), No solution.
2. If P[A:B] = P(A) = the number of unknown variables, unique solution.
3. If P[A:B] = P(A) ≠ number of unknown, infinite number of solutions.

Here P[A:B] is the rank of Gauss Elimination representation of AX = B.
There are two states of the Linear equation system: 

• Consistent State: A System of equations having one or more solutions is called a consistent system of equations.
• Inconsistent State: A System of equations having no solutions is called inconsistent system of equations.

Linear dependence and Linear independence of vector:

Linear Dependence: A set of vectors X₁ ,X₂ ….X_r is said to be linearly dependent if there exist r scalars k₁ ,k₂ …..k_r such that: k₁ X₁ + k₂X₂ +……..k_r X_r = 0. 

Linear Independence: A set of vectors X₁ ,X₂….X_r is said to be linearly independent if for all r scalars k₁,k₂ …..k_r such that k₁X₁+ k₂ X₂+……..k_rX_r = 0, then k₁ = k₂ =……. = k_r = 0. 
How to determine linear dependency and independency ?
Let X₁, X₂ ….X_r be the given vectors. Construct a matrix with the given vectors as its rows. 

1. If the rank of the matrix of the given vectors is less than the number of vectors, then the vectors are linearly dependent.
2. If the rank of the matrix of the given vectors is equal to number of vectors, then the vectors are linearly independent.

Reference: 
http://www.dr-eriksen.no/teaching/GRA6035/2010/lecture2-hand.pdf

This article is contributed by Nitika Bansal.