We take the determinant of both sides of the equation \(MN = mI\):
Since the order \(n=3\) and \(\text{det}(I) = 1\), the right side simplifies to \(\text{det}(mI) = m^3 \cdot 1 = m^3\).
Substitute \(\text{det}(M) = m\) and \(\text{det}(mI) = m^3\) we get :
Given: \(MN = mI\), where M and N are square matrices of order 3, and det(M) = m.
Taking the determinant of both sides of the equation \(MN = mI\):
\(\det(MN) = \det(mI)\)
Using the property that \(\det(AB) = \det(A) \det(B)\):
\(\det(M) \det(N) = \det(mI)\)
Since M and N are of order 3, \(mI\) is a 3x3 matrix with m along the diagonal. Thus,
\(\det(mI) = m^3\)
Substituting the given value \(\det(M) = m\):
\(m \det(N) = m^3\)
Dividing both sides by m (assuming \(m \neq 0\)):
\(\det(N) = \frac{m^3}{m}\)
\(\det(N) = m^2\)
Correct Answer: \(m^{2}\)
AI generated content. Review strictly for academic accuracy.