Show that A+A^T is symmetric for any 2 x 2 matrix A. Is this true for any square matrix?
The correct answer and explanation is:
Correct Answer:
Yes, for any square matrix AA, the matrix A+ATA + A^T is symmetric.
Explanation (300 words):
To determine whether A+ATA + A^T is symmetric, we must recall the definition of a symmetric matrix:
A matrix MM is symmetric if MT=MM^T = M, where MTM^T is the transpose of MM.
Let’s test this with a general 2 × 2 matrix: A=[abcd]A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}
Then, the transpose of AA is: AT=[acbd]A^T = \begin{bmatrix} a & c \\ b & d \\ \end{bmatrix}
Now, compute A+ATA + A^T: A+AT=[abcd]+[acbd]=[a+ab+cc+bd+d]=[2ab+cb+c2d]A + A^T = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} + \begin{bmatrix} a & c \\ b & d \\ \end{bmatrix} = \begin{bmatrix} a + a & b + c \\ c + b & d + d \\ \end{bmatrix} = \begin{bmatrix} 2a & b + c \\ b + c & 2d \\ \end{bmatrix}
This matrix is symmetric because the transpose of A+ATA + A^T is: (A+AT)T=[2ab+cb+c2d]T=[2ab+cb+c2d]=A+AT(A + A^T)^T = \begin{bmatrix} 2a & b + c \\ b + c & 2d \\ \end{bmatrix}^T = \begin{bmatrix} 2a & b + c \\ b + c & 2d \\ \end{bmatrix} = A + A^T
Therefore, A+ATA + A^T is symmetric.
General Case: Any Square Matrix
The above result isn’t limited to 2 × 2 matrices. For any square matrix AA (i.e., of size n×nn \times n), we use the fact that: (A+AT)T=AT+(AT)T=AT+A=A+AT(A + A^T)^T = A^T + (A^T)^T = A^T + A = A + A^T
Since the transpose of A+ATA + A^T equals itself, it is symmetric.
Conclusion:
Yes, A+ATA + A^T is symmetric for any square matrix AA, including 2 × 2 matrices.