find the number of 2*2 matrix satisfying...
i) aij is 1 or -1
ii) (a11)2 +(a12)2 = (a21)2 + (a22)2 =2
iii) a11a21 + a12a22 =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 a11 = a12 = a21 = a22 = 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:
a11 a21 + a12 a22 = 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..