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GATE 2022 EE – Question 42

Engineering Mathematics · Linear Algebra: Matrix Algebra · 2 marks · Multiple choice

Consider a matrix $A=\begin{bmatrix}1&0&0\\0&4&-2\\0&1&1\end{bmatrix}$.

The matrix $A$ satisfies the equation $6A^{-1}=A^2+cA+dI$, where $c$ and $d$ are scalars and $I$ is the identity matrix.

Then $(c+d)$ is equal to

  1. 5
  2. 17
  3. $-6$
  4. 11

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Correct answer: (A) 5

Explanation

The eigenvalues of $A$ are $1$ and the roots of $\lambda^2-5\lambda+6=0$, i.e. $2$ and $3$. The characteristic polynomial is $\lambda^3-6\lambda^2+11\lambda-6$, and by Cayley-Hamilton $A^3-6A^2+11A-6I=0$. Dividing by $A$ gives $6A^{-1}=A^2-6A+11I$, so $c=-6$, $d=11$ and $c+d=5$.