Matrix is given to be symmetric, find the values of a and b.

Engineering

Mathematics

Special Type of Square Matrices

Question

Matrix $A = [\begin{array}{ccc} 0 & 2 b & - 2 \\ 3 & 1 & 3 \\ 3 a & 3 & - 1 \end{array}]$ is given to be symmetric, find the values of a and b.

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$A = [\begin{array}{ccc} 0 & 2 b & - 2 \\ 3 & 1 & 3 \\ 3 a & 3 & - 1 \end{array}]$

$\Rightarrow A^{T} = [\begin{array}{ccc} 0 & 3 & 3 a \\ 2 b & 1 & 3 \\ - 2 & 3 & - 1 \end{array}]$

It is given that given matrix is symmetric.

∴ A = A^T

\Rightarrow [\begin{array}{ccc} 0 & 2 b & - 2 \\ 3 & 1 & 3 \\ 3 a & 3 & - 1 \end{array}] = [\begin{array}{ccc} 0 & 3 & 3 a \\ 2 b & 1 & 3 \\ - 2 & 3 & - 1 \end{array}]

If matrices are equal then there corresponding elements are also equal.

⇒ 2b = 3

\Rightarrow b = \frac{3}{2}

⇒ 3a = – 2

\Rightarrow a = \frac{- 2}{3}