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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 $b$?

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

1. $a = 6, b = 4$
2. $a = 4, b = 6$
3. $a = 3, b = 5$
4. $a = 5, b = 3$

### 1 comment

Trace of any square matrix = sum of eigenvalues.

$1+a=-1+7$

=> $a=5$

$\text{Sum of Eigenvalues} = \text{trace of matrix}$
$\implies -1 + 7 = 1+a$

$\text{Product of Eigenvalues} = \text{Determinant of matrix}$
$\implies -1 \times 7 = a - 4b$

This gives $a = 5$ and $b = 3$

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

sum of eigen values=> trace of the matrix

determinent =>product of eigen values

a+1=6=>a=5

a-4b=-7

5-4b=-7

12=4b

b=3..

so a=5 and b=3
by
Multiplication of Eigen values = Determinant of matrix
-1 * 7 = -7
-7  = a – 4b

now go through options option D fits perfectly so

a = 5
b = 3