Solving Systems of Linear Equations Using Matrices
Homogeneous and non-homogeneous systems of linear equations
A system of equations AX = B is called a homogeneous system if B = O. If B ≠ O, it is called a non-homogeneous system of equations.
e.g., 2x + 5y = 0
3x – 2y = 0
is a homogeneous system of linear equations whereas the system of equations given by
e.g., 2x + 3y = 5
x + y = 2
is a non-homogeneous system of linear equations.
Solution of Non-homogeneous system of linear equations
- Matrix method: If AX = B, then X = A-1B gives a unique solution, provided A is non-singular.
But if A is a singular matrix i.e., if |A| = 0, then the system of equation AX = B may be consistent with infinitely many solutions or it may be inconsistent.
- Rank method for solution of Non-Homogeneous system AX = B
- Write down A, B
- Write the augmented matrix [A : B]
- Reduce the augmented matrix to Echelon form by using elementary row operations.
- Find the number of non-zero rows in A and [A : B] to find the ranks of A and [A : B] respectively.
- If ρ(A) ≠ ρ(A : B) then the system is inconsistent.
- ρ(A) = ρ(A : B) = the number of unknowns, then the system has a unique solution.
- ρ(A) = ρ(A : B) < number of unknowns, then the system has an infinite number of solutions.
Solutions of a homogeneous system of linear equations
Let AX = O be a homogeneous system of 3 linear equations in 3 unknowns.
- Write the given system of equations in the form AX = O and write A.
- Find |A|.
- If |A| ≠ 0, then the system is consistent and x = y = z = 0 is the unique solution.
- If |A| = 0, then the systems of equations has infinitely many solutions. In order to find that put z = k (any real number) and solve any two equations for x and y so obtained with z = k give a solution of the given system of equations.
Consistency of a system of linear equation AX = B, where A is a square matrix
In system of linear equations AX = B, A = (aij)n×n is said to be
- Consistent (with unique solution) if |A| ≠ 0.
i.e., if A is non-singular matrix.
- Inconsistent (It has no solution) if |A| = 0 and (adj A)B is a non-null matrix.
- Consistent (with infinitely m any solutions) if |A| = 0 and (adj A)B is a null matrix.
Rank of matrix
Let A be a m×n matrix. If we retain any r rows and r columns of A we shall have a square sub-matrix of order r. The determinant of the square sub-matrix of order r is called a minor of A order r. Consider any matrix A which is of the order of 3×4 say,
It is 3×4 matrix so we can have minors of order 3, 2 or 1. Taking any three rows and three columns minor of order three. Hence minor of order
Making two zeros and expanding above minor is zero. Similarly we can consider any other minor of order 3 and it can be shown to be zero. Minor of order 2 is obtained by taking any two rows and any two columns.
Minor of order .
Minor of order 1 is every element of the matrix.
Rank of a matrix: The rank of a given matrix A is said to be r if
- Every minor of A of order r+1 is zero.
- There is at least one minor of A of order r which does not vanish. Here we can also say that the rank of a matrix A is said to be r ,if
- Every square submatrix of order r+1 is singular.
- There is at least one square submatrix of order r which is non-singular.
The rank r of matrix A is written as ρ(A) = r.
Echelon form of a matrix
A matrix A is said to be in Echelon form if either A is the null matrix or A satisfies the following conditions:
- Every non- zero row in A precedes every zero row.
- The number of zeros before the first non-zero element in a row is less than the number of such zeros in the next row.
If can be easily proved that the rank of a matrix in Echelon form is equal to the number of non-zero row of the matrix.
Rank of a matrix in Echelon form: The rank of a matrix in Echelon form is equal to the number of non-zero rows in that matrix.
Solving Systems of Linear Equations Using Matrices Problems with Solutions