Understanding Symmetric Matrices

A symmetric matrix is a square matrix that is equal to its transpose. In simpler terms, if A is a symmetric matrix, then \(A = A^T\). This property means that the element in the i-th row and j-th column is equal to the element in the j-th row and i-th column for all i and j. Mathematically, this is expressed as \(a_{ij} = a_{ji}\) for all \(i, j\).

The given matrix is:

\( A = \left[ {\begin{array}{*{20}{c}} 1&{3 + x}&2\\ {1 - x}&2&{y + 1}\\ 2&{5 - y}&3 \end{array}} \right] \)

For this matrix A to be a symmetric matrix, the elements must satisfy the condition \(a_{ij} = a_{ji}\).

Applying the Symmetric Matrix Property

We need to compare the off-diagonal elements of the matrix A based on the property of a symmetric matrix:

• The element in the first row, second column (\(a_{12}\)) must be equal to the element in the second row, first column (\(a_{21}\)).
• The element in the first row, third column (\(a_{13}\)) must be equal to the element in the third row, first column (\(a_{31}\)).
• The element in the second row, third column (\(a_{23}\)) must be equal to the element in the third row, second column (\(a_{32}\)).

Setting up Equations to Find x and y

Let's write down the equations based on the elements of matrix A:

• From \(a_{12} = a_{21}\):
  \(3 + x = 1 - x\)
• From \(a_{13} = a_{31}\):
  \(2 = 2\) (This equation is always true and does not help in finding x or y)
• From \(a_{23} = a_{32}\):
  \(y + 1 = 5 - y\)

Solving for the Variables x and y

Now, we solve the equations we obtained:

Equation 1: \(3 + x = 1 - x\)

Add x to both sides:
\(3 + x + x = 1 - x + x\)
\(3 + 2x = 1\)

Subtract 3 from both sides:
\(2x = 1 - 3\)
\(2x = -2\)

Divide by 2:
\(x = \frac{-2}{2}\)
\(x = -1\)

Equation 2: \(y + 1 = 5 - y\)

Add y to both sides:
\(y + 1 + y = 5 - y + y\)
\(2y + 1 = 5\)

Subtract 1 from both sides:
\(2y = 5 - 1\)
\(2y = 4\)

Divide by 2:
\(y = \frac{4}{2}\)
\(y = 2\)

So, we found the values of x and y are \(x = -1\) and \(y = 2\). This uses the concept of a symmetric matrix.

Calculating 3x + y

The question asks for the value of \(3x + y\). We substitute the values of x and y we just found:

\(3x + y = 3(-1) + 2\)

Perform the multiplication:
\(3(-1) = -3\)

Now add 2:
\(-3 + 2 = -1\)

Thus, the value of \(3x + y\) is -1.

This calculation confirms how the definition of a symmetric matrix helps determine the unknown elements and solve related expressions.

Conclusion

Based on the properties of a symmetric matrix, we found \(x = -1\) and \(y = 2\). Using these values, \(3x + y = -1\).