Question

 If $A=\left[ \begin{matrix} 4 & x+2 \\ 2x-3 & x+1 \\ \end{matrix} \right]$ is symmetric, then $x =$ [Karnataka CET 1994]
A. $3$
B. $5$
C. $2$
D. $4$

Answer 1

Hint:
Here we need to use the property of a symmetric matrix, which is a square matrix equal to its transpose. To get the value of unknown, compare the corresponding elements in the first and second matrices.



Formula Used:
Symmetric Matrix is given by:
$A=A^T$

Complete step-by-step solution:
Given: $A=\left[ \begin{matrix} 4 & x+2 \\ 2x-3 & x+1 \\ \end{matrix} \right]$
And the given matrix $A$ is a symmetric matrix,
$\therefore A=A^T$
Transpose of $A$ is given by;
$A^T=\left[ \begin{matrix} 4 & 2x-3 \\ x+2 & x+1 \\ \end{matrix} \right]$
According to the definition of the symmetric matrix we have;
$\left[ \begin{matrix} 4 & x+2 \\ 2x-3 & x+1 \\ \end{matrix} \right]=
\left[ \begin{matrix} 4 & 2x-3 \\ x+2 & x+1 \\ \end{matrix} \right]$
Both are equal matrices so their corresponding elements are equal.
$\Rightarrow x+2=2x-3\\
\Rightarrow 2x-x=3+2\\
\Rightarrow x=5$
Hence, the value of $x$ is $5$.

So, option is B correct.

Note:
If and only if changing the indices does not alter the elements of matrix $A$, then it is symmetric. Only square matrices can be symmetrical because equal matrices have equal dimensions.

Additional Information:
Symmetric Matrix Properties
The key characteristics of symmetric matrices that set them apart from other kinds of matrices are listed as,
1. The sum of two symmetric matrices is a symmetric matrix.
2. When two symmetric matrices $A$ and $B$ are multiplied, the product matrix $AB$ is symmetric if and only if the two matrices are commutative, that is, if $AB = BA$.
3. For any integer $n$, if A is symmetric, $A^n$ is also symmetric.
4. If a matrix $A$ inverse exists, it will only be symmetric if the square matrix $A$ is symmetric.