$1+a=-1+7$

=> $a=5$

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24 votes

Consider the following $2 \times 2$ matrix $A$ where two elements are unknown and are marked by $a$ and $b$. The eigenvalues of this matrix are $-1$ and $7.$ What are the values of $a$ and $b$?

$\qquad A = \begin{pmatrix}1 & 4\\ b&a \end{pmatrix}$

- $a = 6, b = 4$
- $a = 4, b = 6$
- $a = 3, b = 5$
- $a = 5, b = 3 $

10 votes

As, we know - product of eigenvalues of a matrix is equal to determinant of that matrix.

so,

(-1) × 7

= det(A)

=(1 × a ) - (4 × b)

so, a - 4b = -7 -------(1)

also, trace (sum of the diagonal elements) of a matrix is equal to sum of eigenvalues of the matrix.

so, 1 + a = -1+7 =6

so, a = 5 -------(2)

from equation (1) and (2)

b = 3

So,answer is D