find the number of 2*2 matrix satisfying...

i) a_{ij }is 1 or -1

ii) (a_{11})^{2 }+(a_{12})^{2} = (a_{21})^{2} + (a_{22})^{2} =2

iii) a_{11}a_{21} + a_{12}a_{22} =0

A 2×2 matrix is generally represented as

(i) The condition is aij is 1 or – 1. i.e. each element has two choices.

Total number of elements are 4.

∴ Number of matrix satisfying the condition aij = 1 or – 1 are (2)^{4} = 16

(ii) Given condition is –

In your query, you did not mention from where the entries are taken in the matrix i.e. either aij ∈N or aij ∈ R.

if aij ∈ N, then

is possible only when *a*_{11} = *a*_{12} = *a*_{21} = *a*_{22} = 1.

Hence, the number of 2 × 2 matrix satisfying the condition is only one.

If aij ∈ R, then there will be infinite matrices which satisfies this condition.

(iii) The given condition is:

*a*_{11} *a*_{21} + *a*_{12} *a*_{22} = 0

If aij ∈ R, then this condition is satisfied by infinite values of aij.

Hence, no.of matrix satisfying condition (iii) are infinite.

If aij ∈N, then there will be no matrix which will satisfy this condition as for natural numbers this condition can never hold..

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