Find a, b, c if [ ccc 1 & 3 5 & a b & -5 & -7 -4 & c & 0 ] is a symetric matrix.

Let A = [ ccc 1 & 3 5 & a b & -5 & -7 -4 & c & 0 ] Since, A is a symmetric matrix, a _ ij = a _ ji for all i and j a_ 13 =a_ 31 , a_ 12 =a_ 21 and a_ 23 =a_ 32 a=-4, 3 5 =b and -7=c a=-4, b=3 5 and c=-7. Alternative Method : Let A= ( rrr 1 & 3 & a b & -5 & -7 -4 & c & 0 ) Then A^ T = ( rrr 1 & b & -4 3 & -5 & -7 5 & -7 & 0 ) Since, A is symmetric matrix, A=A^T ( rrr 1 & 3 5 & a b & -5 & -7 -4 & c & 0 )= ( rrr 1 & b & -4 3 & -5 & -7 5 & -7 & 0 ) By equality of matrices a=-4, b=3 5 and c=-7.

Find a, b, c if [ ccc 1 & 3 5 & a b & -5 & -7 -4 & c & 0 ] is a s…

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Question

Find $a, b, c$ if $\left[\begin{array}{ccc}1 & \frac{3}{5} & a \\ b & -5 & -7 \\ -4 & c & 0\end{array}\right]$ is a symetric matrix.

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Answer

Let $A =\left[\begin{array}{ccc}1 & \frac{3}{5} & a \\ b & -5 & -7 \\ -4 & c & 0\end{array}\right]$
Since, $A$ is a symmetric matrix, $a _{ ij }= a _{ ji }$ for all $i$ and $j$
$\therefore a_{13}=a_{31}, a_{12}=a_{21}$ and $a_{23}=a_{32}$
$\therefore a=-4, \frac{3}{5}=b$ and $-7=c$
$\therefore a=-4, b=\frac{3}{5}$ and $c=-7$.
Alternative Method :
Let $A=\left(\begin{array}{rrr}1 & 3 & a \\ b & -5 & -7 \\ -4 & c & 0\end{array}\right)$
Then $A^{ T }=\left(\begin{array}{rrr}1 & b & -4 \\ 3 & -5 & -7 \\ 5 & -7 & 0\end{array}\right)$
Since, $A$ is symmetric matrix, $A=A^T$
$
\therefore\left(\begin{array}{rrr}
1 & \frac{3}{5} & a \\
b & -5 & -7 \\
-4 & c & 0
\end{array}\right)=\left(\begin{array}{rrr}
1 & b & -4 \\
3 & -5 & -7 \\
5 & -7 & 0
\end{array}\right)
$
By equality of matrices
$a=-4, b=\frac{3}{5}$ and $c=-7$.