Question

Prove that rank of a non-Singular matrix is equal to rank of its reciprocal matrix​

Answer 1

refer the above image....

Answer 2

SOLUTION

TO PROVE

The rank of a non-Singular matrix is equal to rank of its reciprocal matrix

PROOF

Let A be a non zero matrix of order m × n.

The Rank of A is defined to be the greatest positive integer r such that A has at least one non-zero minor of order r

For a non-zero m × n matrix A

0 < rank of A ≤ min {m, n}

For a non-zero matrix A of order n,

rank of A < , or = n according as A is singular or non-singular

Since the matrix A and its reciprocal matrix have the identical minors

So the rank of a non-Singular matrix is equal to rank of its reciprocal matrix

Hence proved

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