Mathematics | System of Linear Equations

 

Trace of a matrix :
Let A=[aij] 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+ ……….+ ann

Properties of trace of matrix:
Let A and B be any two square matrix 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 mxn 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 In 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 < 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 Am×1 is a non zero column matrix and B1×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-homogenous 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 rank of gauss elimination representation of AX = B.
There are two states of 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 ….Xr is said to be linearly dependent if there exist r scalars k1 ,k2 …..kr such that: k1 X1 + k2X2 +……..kr Xr = 0.

Linear Independence: A set of vectors X1 ,X2….Xr is said to be linearly dependent if there exist r scalars k1,k2 …..krsuch that: k1X1+ k2 X2+……..krXr = 0 implies k1 = k2 =……. = kr = 0.
How to determine linear dependency and independency ??
Let X1, X2 ….Xr 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.

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