Paragraph: Let p be an odd prime number and T p be the following set of 2 × 2 matrices T p = { A = [ a c b a ] ; a , b , c ∈ { 0 , 1 , 2 , … , p − 1 } } Question: The number of A in T p such

Question

Paragraph: Let p be an odd prime number and T p ​ be the following set of 2 × 2 matrices T p ​ = { A = [ a c ​ b a ​ ] ; a , b , c ∈ { 0 , 1 , 2 , … , p − 1 } } Question: The number of A in T p ​ such that the trace of A is not divisible by p but det ( A ) is divisible by p is [Note : The trace of a matrix is the sum of its diagonal entries.]

TestHub · Accepted Answer

Trace of A = 2 a , will be divisible by p iff a = 0 . ∣ A ∣ = a 2 − b c , for ( a 2 − b c ) to be divisible by p . There are exactly ( p − 1 ) ordered pairs ( b , c ) for any value of a . ∴ Required number is ( p − 1 ) 2 . Hence, (c) is the correct option.

Mathematics - Matrices Question with Solution | TestHub

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