Summary

Summary

In this chapter we have acquired the knowledge of

·           Rank of a matrix

The rank of a matrix A is the order of the largest non-zero minor of A

·           The rank of a matrix A is the order of the largest non-zero minor of A

·           ρ (A) ≥ 0

·           If A is a matrix of order m × n , then ρ(A) ≤ minimum of {m,n}

·           The rank of a zero matrix is ‘0’

·           The rank of a non- singular matrix of order n × n is ‘n’

·           Equivalent Matrices

Two Matrices A and B are said to be equivalent if one can be obtained from another by a finite number of elementary transformations and we write it as A ~ B .

·           Echelon form

A matrix of order m × n is said to be in echelon form if the row having all its entries zero will lie below the row having non-zero entry.

·           A system of equations is said to be consistent if it has at least one set of solution. Otherwise, it is said to be inconsistent

If ρ ( [A, B] ) = ρ ( A) , then the equations are consistent.

If ρ ( [A, B] ) = ρ ( A)= n , then the equations are consistent and have unique solution.

If ρ ( [A, B] ) = ρ ( A) < n , then the equations are consistent and have infinitely many solutions.

If ρ ([A, B] ) ≠ ρ ( A) then the equations are inconsistent and has no solution.

·           | adjA| = |A|ⁿ⁻¹

·           If |A| = 0 then A is a singular matrix. Otherwise, A is a non singular matrix.

·           In AX = B if |A| ≠ 0 then the system is consistent and it has unique solution.

·           Cramer’s rule is applicable only when Δ ≠ 0 .