Matrices & Determinants
Skew-Symmetric — B^T B and Trace
nta_pyq_2026_jan
Grade 12
Question:
Let $A=\begin{bmatrix}0&2&-3\\-2&0&1\\3&-1&0\end{bmatrix}$ and $B$ be a matrix such that $B(I-A)=I+A$. Then the sum of the diagonal elements of $B^TB$ is equal to _____.
Step-by-Step Solution
Key Concept: $B=(I+A)(I-A)^{-1}$. Since $A$ is skew-symmetric, $(I-A)$ and $(I+A)$ commute. $B^TB=(I+A)^{-1}(I-A)(I+A)(I-A)^{-1}=(I+A)^{-1}(I+A)(I-A)(I-A)^{-1}=I$.
$B^TB=I$. Sum of diagonal elements $=3$.
Correct Answer: 3