Mathematics Olympiad ~ Vinod Sing, Kolkata
$Problem$ #1 $$ $$
If $A$ and $B$ are different matrices satisfying \( A^3 = B^3 \) and \(A^2B = B^2A\), find \(det(A^2+B^2)\) $$ $$
Since $A$ and $B$ are different matrices \( A-B \neq O \), Now \((A^2+B^2)(A-B) = A^3-A^2B+B^2A-B^3\) $$ $$
=$O$ since \(A^3 = B^3\) and \(A^2B = B^2A\) $$ $$
This shows that \((A^2+B^2)\) has a zero divisor, so it is not invertible hence \(det(A^2+B^2) = 0\)
No comments:
Post a Comment