Linear System of Equations and Determinants

Linear system of m equations in n unknowns x_1, x_2, …, x_n  is the set of equations
          a₁₁x₁ + a₁₂x₂ + … + a_1nx_n = b₁
             a₂₁x₁ + a₂₂x₂ + … + a_2nx_n = b_{2
             .
             .
             .}
             a_m1x₁ + a_m2x₂ + … + a_mnx_n = b_m
here a_ij are called coefficients(which are given numbers)

1. If all b_j  (j = 1, 2, ..., m) are zero then homogeneous system.
2. If atleast one b_j  (j = 1, 2, ..., m) is not zero then non-homogeneous.
3. Solution- set of numbers x_1, x_2, …, x_n which satisfies all m- equations.
4. Solution vector- Ordered Set of numbers [x_1, x_2, …, x_n] which satisfies all m- equations.
5. If the above is homogeneous system, then it has atleast one trivial solution.

              x₁ =x₂ = x₃ = … = x_n = 0

Rank of Matrix: Linear Independence and Dependence
Let a̅_1, a̅_{2, ...,} a̅_m are m- vectors, then
Linear combination of m-vectors: 
c₁a̅₁ + c₂a̅₂ + … + c_ma̅_m = ∑c_ia̅_i  ∀ i= 1, 2,..., m 
where, c_1, c_{2, ...,} c_m are any scalars.

Linearly independent vectors- If  ∑c_ia̅_i  
∀ i= 1, 2,..., m = 0̅, when all c_i's are zero. then
( a̅_1, a̅_{2, ...,} a̅_m) are linearly independent vectors.

Linearly dependent vectors-  
If  ∑c_ia̅_i  ∀ i= 1, 2,..., m = 0̅, for some c_i's may be zero, then
( a̅_1, a̅_{2, ...,} a̅_m) are linearly dependent vectors.

Sub-matrix- A matrix obtained from a matrix, by omitting rows and columns.

Rank of a matrix- The maximum number of linearly independent row vectors of a matrix
  A = [a_jk] is called the rank of A or (rank A).

Nullity of a matrix-  If A is a square matrix of order n  then nullity of matrix A,
                 N(A) = n - rank A.

Some related theorems:

1. The rank of a matrix A equals the maximum number of linearly independent columns vectors of A.
2. Matrix A and its transpose A^T have same rank.
3. Row-equivalent matrix have the same rank.
4. Rank of the product of two matrices cannot exceed the rank of either factor.

Determinants:
The n-order determinant of a square matrix A = [a_ij] of order n, is a number,
                  det. A = |A| = | a_ij |
                                        = a_j1c_j1 + … + a_jnc_jn , j = 1,2,… or n
                                        = a_1kc_1k + … + a_nkc_nk , k = 1,2,… or n
where, c_jk   = (-1)^j+k M_{jk
                         = Co-factor of}  a_ik in |A|
and,      m_{jk = Determinant of order (n - 1), obtained by deleting the rows and columns of entry} a_{jk
                       (i.e., j-th row and k-th column).
                            = minor of}  a_jk in |A|.
Geometrically,
          det A = ± volume of the n-dimensional parallelopiped spanned by the column (or row) vectors of A.

Some properties of Determinants: Let A be a determinant of order n.

1. det A^T= det A
2. |AB| = |A| |B|
3. det A^-1= 1/det A
4. det I = 1
5. det (xA) = xⁿdet A
6. det A = λ₁ . . . λ_n = product of eigen values.
7. If all elements of a row (or column) are multiplied by constant k, then determinant is multiplied by k.
8. Exchange of two rows (or columns) change the sign of the determinant.
9. Determinant does not change if one row (or columns) multiplied by a constant is added to another row (or column)
10. Determinant equals to zero, if
    (a) All elements of a row (column) are zero, or
    (d) Two rows (columns) coincide.

Rank of a matrix in terms of determinant- An m × n matrix A = [a_ij] has rank r ≥ 1 iff A has r × r sub matrix with non-zero determinant.

If A is square matrix of order n, its rank is a iff det A ≠ 0.

Cramer's Theorem (Solution of linear system by determinants):
(a) If a linear system of n-equations has the same number of unknowns x₁ , . . . ,  x_n
            a₁₁x₁ + a₁₂x₂ + … + a_1nx_n = b₁
              a₂₁x₁ + a₂₂x₂ + … + a_2nx_n = b_{2
             .
             .
             .}
             a_n1x₁ + a_n2x₂ + … + a_nnx_n = b_n

                      ⇔ A̅ x̅ = b̅
has a non-zero coefficient determinant D = det A, the system  has precisely one solution. This solution is given by the formulas
           x₁ = D₁/D ,  x₂ = D₂/D, . . . , x_n = D_n/D (Cramer's Rule)
where D_k is the determinant obtained from D by replacing in D the k-th column by the column with entries b₁ , . . . ,  b_n
If the system is homogeneous and D ≠ 0, then it has only the trivial solution x₁ = 0 , . . . ,  x_n = 0.
If D = 0, the homogenoeous system also hace non-trivial solutions.