Question

Suppose \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + qx + r = 0 \) (with \( r \neq 0 \)) and they are in A.P. Then the rank of the matrix \( \begin{pmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{pmatrix} \) is:

• A. 3
• B. 2
• C. 0
• D. 1

Hint:

For matrix rank, use row reduction or determinant. For roots in A.P., express them symmetrically and check linear dependence in the matrix.

Answer 1

The Correct Option is B

To determine the rank of the matrix \(\begin{pmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{pmatrix}\), where \( \alpha, \beta, \gamma \) are roots of the polynomial \( x^3 + qx + r = 0 \) and are in an arithmetic progression (A.P.), we proceed as follows:

Since \( \alpha, \beta, \gamma \) are in A.P., we express them as:
\(\beta = \alpha + d\),
\(\gamma = \alpha + 2d\).

Thus, the roots are \(\alpha, \alpha + d, \alpha + 2d\).

By Viète's formulas, the sum of the roots of the cubic equation is:
\(\alpha + \beta + \gamma = 0\).
Substituting values:
\(\alpha + (\alpha + d) + (\alpha + 2d) = 0\)
\(3\alpha + 3d = 0\)
\(\alpha + d = 0\)
\(d = -\alpha\).

Therefore, \(\beta = \alpha - \alpha = 0\) and \(\gamma = \alpha + 2(-\alpha) = -\alpha\).

Substituting into the matrix:

\(\alpha\)	0	-\(\alpha\)
0	-\(\alpha\)	\(\alpha\)
-\(\alpha\)	\(\alpha\)	0

The rows and columns are linearly dependent. Hence, the rows and columns are not fully independent.

To confirm non-full rank, calculate the determinant:

\(\text{Let } A = \begin{pmatrix} \alpha & 0 & -\alpha \\ 0 & -\alpha & \alpha \\ -\alpha & \alpha & 0 \end{pmatrix}\).
The determinant of \( A \):
\(\det(A) = \alpha(0\cdot0 - (-\alpha)\cdot\alpha) - 0 + (-\alpha)(0\cdot0 - (-\alpha)\cdot\alpha)\)
\(= \alpha^3 + \alpha^3 = 2\alpha^3\),
which is \(0\) only if \(\alpha = 0\), contradicting \(r \neq 0\).

A non-zero determinant implies potential rank issues. The rank is no higher than 2 due to linear relationships between columns.

Therefore, the rank of the matrix is 2.