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Husk, 12 bytes

S≡o ́Ṫ§oε%=▼l·

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Explanation

S≡o ́Ṫ§oε%=▼l· Input is a k×ばつk array A.
 l· The range [1,2..k]
 o ́Ṫ Outer product with itself by this function:
 Arguments are two numbers x and y.
 ▼ Minimum of x and y
 §% modulo
 = equality of x and y.
 oε Is it at most 1?
S≡ Does the result have the same shape and distribution of truthy values as A?