If \({\rm{A}} = \left( {\begin{array}{*{20}{c}} 4&{{\rm{x}} + 2}\\ {2{\rm{x}} - 3}&{{\rm{x}} + 1} \end{array}} \right)\) is symmetric, then what is x equal to?

Concept:

Symmetric Matrix: If the transpose of a matrix is equal to itself, that matrix is said to be symmetric.

Or, A matrix A is symmetric if and only if swapping indices doesn't change its components

• A = A^T
• a_ij = a_ji

CALCULATION:

Given - \({\rm{A}} = \left( {\begin{array}{*{20}{c}} 4&{{\rm{x}} + 2}\\ {2{\rm{x}} - 3}&{{\rm{x}} + 1} \end{array}} \right)\)

A real square matrix A = (a_ij) is said to be symmetric, if A = A^T

Where A^T = transpose of matrix A

\({{\rm{A}}^{\rm{T}}} = \left( {\begin{array}{*{20}{c}} 4&{2{\rm{x}} - 3}\\ {{\rm{x}} + 2}&{{\rm{x}} + 1} \end{array}} \right)\)

∴ A = A^T

\(\Rightarrow \left[ {\begin{array}{*{20}{c}} 4&{{\rm{x}} + 2}\\ {2{\rm{x}} - 3}&{{\rm{x}} + 1} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 4&{2{\rm{x}} - 3}\\ {{\rm{x}} + 2}&{{\rm{x}} + 1} \end{array}} \right]\)

Compare A₂₁ element.

⇒ x + 2 =2x - 3 

⇒ x = 5