QS-024 · Linear Algebra · Easy
NPTEL GATE DA MOCK | Week 1 - 58
How many orthogonal matrices of order $n$ exist that are made up of only zeros and ones?
A. $n^2$
B. $2^n$
C. $n!$
D. $n$
How many orthogonal matrices of order $n$ exist that are made up of only zeros and ones?
A. $n^2$
B. $2^n$
C. $n!$
D. $n$
Solution
For an orthogonal matrix $A$,
$$ A^TA=I. $$
Since $A$ contains only 0 and 1, each column must have exactly one 1. Also, no two columns can have their 1 in the same row.
Thus, every row and every column contains exactly one 1. Such a matrix is a permutation matrix.
The number of permutation matrices of order $n$ equals the number of permutations of $n$ elements:
$$ n!=n(n-1)(n-2)\cdots1. $$
Therefore, n!
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