If \( A, B \), and \( C \) are the angles of a triangle and \[ \begin{vmatrix} 1 & 1 & 1 \\ 1 + \sin A & 1 + \sin B & 1 + \sin C \\ \sin A + \sin^2 A & \sin B + \sin^2 B & \sin C + \sin^2 C \end{vmatrix} = 0, \] then which of the following is correct?
The triangle ABC is isosceles
To solve this problem, we need to evaluate the determinant given:
| 1 | 1 | 1 |
| 1 + \(\sin A\) | 1 + \(\sin B\) | 1 + \(\sin C\) |
| \(\sin A + \sin^2 A\) | \(\sin B + \sin^2 B\) | \(\sin C + \sin^2 C\) |
The determinant of the matrix is set to 0:
\(\begin{vmatrix} 1 & 1 & 1 \\ 1 + \sin A & 1 + \sin B & 1 + \sin C \\ \sin A + \sin^2 A & \sin B + \sin^2 B & \sin C + \sin^2 C \end{vmatrix} = 0\)
This means the rows of the matrix are linearly dependent. Without loss of generality, consider a simpler matrix where simple symmetry helps us deduce properties about the angles:
The dependence implies a relation between the angles. Specifically, if two angles, say \(A\) and \(B\), are equal, then the corresponding sine terms will ensure that the determinant remains zero due to the dependency aligning in symmetry.
Thus, in geometrical terms, if two angles are equal, that leads the triangle to be isosceles.
Hence, based on the equality conditions and determinant properties, we conclude:
The triangle \(ABC\) is isosceles.
Important Questions from Determinants
If \(A=\left[\begin{array}{rrr} 2 & -1 & 0 \\ -1 & 3 & 0 \\ 1 & 0 & 1 \end{array}\right]\), then what is the value of det[adj(adjA)] ?
If A, B and C are square matrices of order 3 and det(BC) = 2 det(A), then what is the value of det(2A-1BC)?
If \(A=\left[\begin{array}{rrr} 0 & 3 & 4 \\ -3 & 0 & 5 \\ -4 & -5 & 0 \end{array}\right]\), then which one of the following statements is correct?
If \(\left|\begin{array}{ccc} x^2+3 x & x-1 & x+3 \\ x+1 & -2 x & x-4 \\ x-3 & x+4 & 3 x \end{array}\right|\) = ax4 + bx3 + cx2 + dx + e, then what is the value of e?"
If all elements of a third order determinant are equal to 1 or -1, then the value of the determinant is:
