Question

If A is a square matrix, then which of the following is true for \[A + {A^T}\]?
A. Nonsingular matrix
B. Symmetric matrix
C. Skew-symmetric matrix
D. Unit matrix

Answer 1

Hint: To solve this question, we check whether the transpose of the given matrix is a symmetric matric or skew-symmetric matrix. To check it, we will take find the transpose of \[A + {A^T}\] and apply the transpose of the sum of two matrices. Then apply transpose of transpose a matrix and commutative property to get the required answer.

Formula Used:
The transpose of the given matrix is the given matrix.
\[{\left( {{A^T}} \right)^T} = A\]
Transpose of the sum of two matrices:
\[{\left( {A + B} \right)^T} = {A^T} + {B^T}\]

Complete step by step solution:
Given that A is a square matrix. Given matrix is \[A + {A^T}\].
Now we will find the transpose of the given matrix
\[{\left( {A + {A^T}} \right)^T}\]
Now apply the transpose of the sum of two matrices:
\[ = {\left( A \right)^T} + {\left( {{A^T}} \right)^T}\]
Now applying the transpose of a matrix
\[ = {\left( A \right)^T} + A\]
The sum of matrices follows the commutative property
\[ = A + {\left( A \right)^T}\]
Since \[{\left( {A + {A^T}} \right)^T} = A + {A^T}\], it is a symmetric matrix.
Hence option B is the correct option.

Additional information:
The sum of a matrix with the transpose of the matrix is possible when the matrix is a square matrix. The symmetric property and skew-symmetric property are applicable to a square matrix.
In the transpose matrix, we interchange the rows into columns or columns into rows.

Note: Students are often confused about the sum of matrices and the multiplication of matrices. The sum of matrices follows the commutative property. But the multiplication of two matrices does not follow the commutative property. Thus we can use the commutative property in the \[ {\left( A \right)^T} + A\] to get final answer.