Question

Let \( A = \begin{bmatrix} b^2+c^2 & a^2 & a^2 \\ b^2 & c^2+a^2 & b^2 \\ c^2 & c^2 & a^2+b^2 \end{bmatrix} \). If \(a = \sin \pi/6, b = \cos \pi/4\) and \(c = \cot \pi/2\) then A is

• A. Symmetric matrix
  % Telugu: సౌష్టవ మాత్రిక
• B. Skew-Symmetric matrix
  % Telugu: అష్టపది మాత్రిక (likely means Skew-Symmetric or similar)
• C. Singular matrix
  % Telugu: అసాధారణ మాత్రిక
• D. Non-singular matrix % Telugu: సాధారణ మాత్రిక

Hint:

A matrix A is singular if its determinant, det(A), is 0. Otherwise, it is non-singular.
Symmetric matrix: \(A = A^T\).
Skew-symmetric matrix: \(A = -A^T\) (implies diagonal elements are 0).
Calculate trigonometric values: \(\sin(\pi/6)=1/2\), \(\cos(\pi/4)=1/\sqrt{2}\), \(\cot(\pi/2)=0\).

Answer 1

The Correct Option is C

First, calculate the values of a, b, and c: \(a = \sin(\pi/6) = \sin(30^\circ) = 1/2\). \(b = \cos(\pi/4) = \cos(45^\circ) = 1/\sqrt{2}\). \(c = \cot(\pi/2) = \cot(90^\circ) = \frac{\cos(90^\circ)}{\sin(90^\circ)} = \frac{0}{1} = 0\). Now, calculate \(a^2, b^2, c^2\): \(a^2 = (1/2)^2 = 1/4\). \(b^2 = (1/\sqrt{2})^2 = 1/2\). \(c^2 = (0)^2 = 0\). Substitute these into matrix A: \(A = \begin{bmatrix} b^2+c^2 & a^2 & a^2
b^2 & c^2+a^2 & b^2
c^2 & c^2 & a^2+b^2 \end{bmatrix} = \begin{bmatrix} 1/2+0 & 1/4 & 1/4
1/2 & 0+1/4 & 1/2
0 & 0 & 1/4+1/2 \end{bmatrix}\) \[ A = \begin{bmatrix} 1/2 & 1/4 & 1/4
1/2 & 1/4 & 1/2
0 & 0 & 3/4 \end{bmatrix} \] A matrix is singular if its determinant is 0. Let's calculate det(A): \(\det(A) = (1/2) \begin{vmatrix} 1/4 & 1/2
0 & 3/4 \end{vmatrix} - (1/4) \begin{vmatrix} 1/2 & 1/2
0 & 3/4 \end{vmatrix} + (1/4) \begin{vmatrix} 1/2 & 1/4
0 & 0 \end{vmatrix}\) \(= (1/2) ((1/4)(3/4) - (1/2)(0)) - (1/4) ((1/2)(3/4) - (1/2)(0)) + (1/4) ((1/2)(0) - (1/4)(0))\) \(= (1/2) (3/16 - 0) - (1/4) (3/8 - 0) + (1/4) (0)\) \(= (1/2)(3/16) - (1/4)(3/8) + 0\) \(= 3/32 - 3/32 = 0\). Since \(\det(A) = 0\), the matrix A is a singular matrix. This matches option (c). Is it symmetric? \(A^T = \begin{bmatrix} 1/2 & 1/2 & 0
1/4 & 1/4 & 0
1/4 & 1/2 & 3/4 \end{bmatrix}\). \(A \neq A^T\). Not symmetric. Not skew-symmetric (diagonal elements not all zero). \[ \boxed{\text{Singular matrix}} \]