Question

If the matrix \(A=\begin{bmatrix} 5&4a+6\\a+12&a+3 \end{bmatrix} \)is a symmetric matrix, then the value of a is:

• A. 2
• B. 5
• C. -2
• D. 3

Answer 1

The Correct Option is A

Given the matrix \(A=\begin{bmatrix} 5 & 4a+6 \\ a+12 & a+3 \end{bmatrix}\). For the matrix to be symmetric, the elements across the main diagonal must be equal, i.e., the matrix \(A\) should satisfy the condition \(A = A^T\). This implies: \[A[i,j] = A[j,i]\] for all \(i\) and \(j\).

The matrix \(A\) is \(\begin{bmatrix} 5 & 4a+6 \\ a+12 & a+3 \end{bmatrix}\).

For symmetry, equate the off-diagonal elements: \[4a + 6 = a + 12\].

To find the value of \(a\), solve the equation:

\[4a + 6 = a + 12\]

Subtract \(a\) from both sides:

\[3a + 6 = 12\]

Subtract 6 from both sides:

\[3a = 6\]

Divide by 3:

\[a = 2\]

Thus, when \(a = 2\), the matrix \(A\) becomes \(\begin{bmatrix} 5 & 14 \\ 14 & 5 \end{bmatrix}\), which is symmetric. Therefore, the value of \(a\) is 2.