Question

सिद्ध कीजिए कि किसी व्युत्क्रमणीय आव्यूह की जाति उसके व्युत्क्रम की जाति के बराबर होती है। in B sc 1st yaar math 1st paper me 1st chapter ​

Answer 1

Given : व्युत्क्रमणीय आव्यूह की कोटि  उसके व्युत्क्रम की कोटि  के बराबर होती है।

Order of inverse matrix is same of order of Matrix

To Find : Prove that

सिद्ध कीजिए

Solution:

Assume that A is the matrix of order m * n  

A⁻¹   is the inverse matrix of A

Let say order of A⁻¹    is  p * q

now we know that

AA⁻¹  =  A⁻¹A  = I

A  order  m * n

A⁻¹  order  p * q

AA⁻¹   multiplication is only possible if

n = p   and   resultant matrix order is  m * q

A⁻¹  order  p * q

A  order  m * n

A⁻¹A multiplication is only possible if

q = m   and  resultant matrix order is  p *  n

As resultant matrix is I

hence  m * q    =  p *  n

=> m = p   and  q = n

q = m  and  m = p  => m = p = q

n = p  and q = n => n  = p = q

Hence m = n = p = q

so order of matrix A and A⁻¹   are m * m

Hence Same

QED

Hence proved

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