Question

If $M$ and $N$ are square matrices of order 3 such that $\det(M) = m$ and $MN = mI$, then $\det(N)$ is equal to :

• A. $-1$
• B. 1
• C. $-m^2$
• D. $m^2$

Hint:

For matrix determinants, use the property $\det(AB) = \det \cdot \det$ and manipulate the equation to solve for unknowns.

Answer 1

The Correct Option is D

We are given that $\det(M) = m$ and $MN = mI$. Taking the determinant of both sides, we get: \[ \det(MN) = \det(mI) \] Since $\det(MN) = \det(M) \cdot \det(N)$ and $\det(mI) = m^3$, we have: \[ \det(M) \cdot \det(N) = m^3 \] Substituting $\det(M) = m$, we get: \[ m \cdot \det(N) = m^3 \quad \Rightarrow \quad \det(N) = m^2 \] Thus, the correct answer is $m^2$.