To determine the value of [tex]\( x \)[/tex] such that the given matrix [tex]\( A \)[/tex] is symmetric, we need to recall the definition of a symmetric matrix. For a matrix to be symmetric, it must be equal to its transpose. In other words, the element at position [tex]\((i, j)\)[/tex] must be equal to the element at position [tex]\((j, i)\)[/tex] for all [tex]\(i\)[/tex] and [tex]\(j\)[/tex].
Given the matrix:
[tex]\[ A = \begin{pmatrix} 4 & x+3 \\ 2x-1 & -1 \end{pmatrix} \][/tex]
For [tex]\( A \)[/tex] to be symmetric, the element in the first row, second column ([tex]\(a_{12}\)[/tex]) must be equal to the element in the second row, first column ([tex]\(a_{21}\)[/tex]). That is:
[tex]\[ a_{12} = a_{21} \][/tex]
Substituting the values from the matrix, we get:
[tex]\[ x + 3 = 2x - 1 \][/tex]
To find [tex]\( x \)[/tex], we solve this equation step-by-step:
1. [tex]\( x + 3 = 2x - 1 \)[/tex]
Subtract [tex]\( x \)[/tex] from both sides to begin isolating [tex]\( x \)[/tex]:
[tex]\[ 3 = x - 1 \][/tex]
Next, add 1 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ 3 + 1 = x \][/tex]
[tex]\[ 4 = x \][/tex]
So, the value of [tex]\( x \)[/tex] that makes the matrix [tex]\( A \)[/tex] symmetric is:
[tex]\[ x = 4 \][/tex]
