To understand the given problem, let's first review the concept of determinants for a 2x2 matrix.

For a 2x2 matrix A = [a b; c d], the determinant is calculated as:

det(A) = ad - bc

In this case, we are given that the determinant of a 2x2 matrix is 50. So we can write the equation as:

det(A) = 50



Eigenvalues of a matrix can be found by solving the characteristic equation, which is obtained by equating the determinant of the matrix minus lambda times the identity matrix to zero.

For a 2x2 matrix A, the characteristic equation is given by:

det(A - λI) = 0

where λ is the eigenvalue and I is the identity matrix.

In this problem, we are given that one eigenvalue is 10. So we can write the equation as:

det(A - 10I) = 0

Now, let's substitute the given determinant value of 50 into this equation and solve for the other eigenvalue.



We have the equation:

det(A - 10I) = 0

Substituting the given determinant value:

det(A - 10I) = 50

Expanding the determinant:

(a - 10)(d - 10) - bc = 50

Since we know that the determinant is equal to 50:

(ad - 10a - 10d + 100) - bc = 50

ad - 10a - 10d + 100 - bc - 50 = 0

ad - 10a - 10d - bc + 50 = 0

Now, let's rearrange the terms and factorize:

(ad - bc) - 10(a + d) + 50 = 0

Since we have the given value of the determinant (ad - bc) as 50:

50 - 10(a + d) + 50 = 0

-10(a + d) + 100 = 0

-10(a + d) = -100

(a + d) = 10

We know that the sum of the eigenvalues is equal to the trace of the matrix, which is the sum of the diagonal elements. In this case, the matrix is 2x2, so the trace is given by (a + d).

Therefore, the sum of the eigenvalues is 10. Since one eigenvalue is given as 10, the other eigenvalue must be the difference between the sum of eigenvalues and the given eigenvalue:

Other eigenvalue = (Sum of eigenvalues) - (Given eigenvalue) = 10 - 10 = 0

Therefore, the other eigenvalue is 0.

However, the correct answer provided is 5, which contradicts the calculated value of 0. It is possible that there may be an error in the given information or a mistake in the calculation.