GATE Overflow Test Series | Linear Algebra | Test 1 | Question: 24

Question

Let $A$ be a real $2 \times 2$ matrix. If $7+5i$ is an eigenvalue of $A$, $\text{det(A)} = \alpha,$ and $\text{trace(A)} = \beta,$ then the value of $2\alpha + 3\beta$ is _______

Answer 1

Eigen values always occur in conjugate pairs. So, if the $\lambda_{1} = 7 + 5i,$ then $\lambda_{2} = 7 - 5i.$

Now, $\text{det(A)} = \lambda_{1} \cdot \lambda_{2} = (7 + 5i)\cdot(7 - 5i) = 49 - 25i^{2} = 49 + 25 = 74.\qquad [\because i^{2} = -1]$

$\implies \text{det(A)} = \alpha = 74.$

And, $\text{trace(A)} = \lambda_{1} + \lambda_{2} = 7 + 5i + 7 - 5i = 14.$

$\implies \text{trace(A)} = \beta = 14.$

Now, the value of $2\alpha + 3 \beta = 2(74) + 3(14) = 148 + 42 = 190.$

So, the correct answer is $190.$

$\text{Important properties of Eigenvalues:}$

1. Sum of all eigenvalues $=$ Sum of leading diagonal(principle diagonal) elements $=$ Trace of the matrix.
2. Product of all eigenvalues $ = \text{det(A)}= \mid A \mid$
3. Any square diagonal(lower triangular or upper triangular) matrix eigenvalues are leading diagonal (principal diagonal) elements itself.

Example$:\;A=\begin{bmatrix} 1& 0& 0\\ 0&1 &0 \\ 0& 0& 1\end{bmatrix}$

• Diagonal matrix, eigenvalues are $1,1,1$

$B=\begin{bmatrix} 1& 9& 6\\ 0&1 &12 \\ 0& 0& 1\end{bmatrix}$

• Upper triangular matrix, eigenvalues are $1,1,1$

$C=\begin{bmatrix} 1& 0& 0\\ 8&1 &0 \\ 2& 3& 1\end{bmatrix}$

• Lower triangular matrix, eigenvalues are $1,1,1$