Q172 / 270 · NDA 2026 (Phase 1) · Mathematics
If $M$ is a square matrix such that $M^3=M$, then how many values of $|M|$ are possible ?
- AOne
- BTwo
- Correct answerCThree
- DFour
Solution
Given $M^3 = M$
Taking determinant on both sides, we get:
$|M^3| = |M|$
Using the property $|A^n| = |A|^n$, we have:
$|M|^3 = |M|$
$|M|^3 - |M| = 0$
$|M|(|M|^2 - 1) = 0$
$|M|(|M| - 1)(|M| + 1) = 0$
$|M| = 0, 1, -1$
Thus, there are three possible values for $|M|$.
Answer: Three
Taking determinant on both sides, we get:
$|M^3| = |M|$
Using the property $|A^n| = |A|^n$, we have:
$|M|^3 = |M|$
$|M|^3 - |M| = 0$
$|M|(|M|^2 - 1) = 0$
$|M|(|M| - 1)(|M| + 1) = 0$
$|M| = 0, 1, -1$
Thus, there are three possible values for $|M|$.
Answer: Three
Hence, the correct answer is option: 'C'.