Matrices & Determinants
Matrix Inverse Relations — x₁+x₂
nta_pyq_2026_jan
Grade 12
Question:
Let $A$, $B$ and $C$ be three $2\times2$ matrices with real entries such that $B=(I+A)^{-1}$ and $A+C=I$. If $BC=\begin{bmatrix}1&-5\\-1&2\end{bmatrix}$ and $CB\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}12\\-6\end{bmatrix}$, then $x_1+x_2$ is
Step-by-Step Solution
Key Concept: $B=(I+A)^{-1}\Rightarrow B(I+A)=I\Rightarrow BA=I-B$. $C=I-A$. $CB=(I-A)B=(I-A)(I+A)^{-1}$. From $BC=2B-I$: $B=\tfrac{1}{2}(BC+I)$.
$x_1=2$, $x_2=-2$. $x_1+x_2=0$.
Correct Answer: 3