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 $MN = mI$. Applying the determinant to both sides yields $\det(MN) = \det(mI)$. Using the properties $\det(MN) = \det(M) \cdot \det(N)$ and $\det(mI) = m^3$, we obtain $\det(M) \cdot \det(N) = m^3$. Substituting $\det(M) = m$ gives $m \cdot \det(N) = m^3$. This simplifies to $\det(N) = m^2$. Therefore, the determinant of $N$ is $m^2$.