Matrices & Determinants
Matrix Transpose Condition — Sum of α Values
nta_pyq_2023_apr
Grade 12
Question:
Let $A$ be a $2\times2$ matrix with real entries such that $A^\top=\alpha A+I$, where $\alpha\in\mathbb{R}\setminus\{-1,1\}$. If $\det(A^2-A)=4$, the sum of all possible values of $\alpha$ is equal to
0
$\dfrac{3}{2}$
2
$\dfrac{5}{2}$
Step-by-Step Solution
Key Concept: Let $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$. From $A^\top=\alpha A+I$: $a=\frac{1}{1-\alpha}$, $d=\frac{1}{1-\alpha}$, $b=c=0$. Then $\det(A^2-A)=\det(A)\det(A-I)$.
$t(t-1)=\pm2$ where $t=\frac{1}{1-\alpha}$. Solutions give $\alpha=\frac{1}{2},2$. Sum $=\frac{5}{2}$.
Correct Answer: 4