IIT-JEE 1982Question 1
Find all values of $\lambda$ such that $(x,y,z)\neq(0,0,0)$ and $$\left(\hat i+\hat j+3\hat k\right)x+\left(3\hat i-3\hat j+\hat k\right)y+\left(-4\hat i+5\hat j\right)z=\lambda\left(x\hat i+y\hat j+z\hat k\right)$$ where $\hat i,\hat j,\hat k$ are unit vectors along the coordinate axes.
Vector Algebra
Answer$\lambda=0,\ -1$
Solution
Need $(M-\lambda I)(x,y,z)^T=0$ with a nonzero solution, $M=\begin{pmatrix}1&3&-4\\1&-3&5\\3&1&0\end{pmatrix}$. $\det(\lambda I-M)=\lambda(\lambda+1)^2=0$, so $\lambda=0$ or $\lambda=-1$.