Matrices & Determinants
A²=I Conditions — Sum of Diagonal and Determinant
nta_pyq_2023_apr
Grade 12
Question:
Let $A=[a_{ij}]_{2\times2}$, where $a_{ij}\neq0$ for all $i,j$ and $A^2=I$. Let $a$ be the sum of all diagonal elements of $A$ and $b=|A|$. Then $3a^2+4b^2$ is equal to
Step-by-Step Solution
Key Concept: $A^2=I$ implies $A$ is an involutory matrix. From $A^2=I$: $a_{12}(a_{11}+a_{22})=0$ and since $a_{12}\neq0$, $a_{11}+a_{22}=0$, giving trace $a=0$. Also $|A|^2=|I|=1\Rightarrow b=\pm1$.
$a=0,\ b=-1$. $3a^2+4b^2=4$.
Correct Answer: 1