Understanding how to express a given matrix as the sum of a symmetric matrix and a skew-symmetric matrix is a fundamental concept in linear algebra. Any square matrix \(A\) can be uniquely represented as the sum of a symmetric matrix \(P\) and a skew-symmetric matrix \(Q\).

The general formulas for \(P\) and \(Q\) are:

• The symmetric part: \(P = \frac{1}{2}(A + A^T)\)
• The skew-symmetric part: \(Q = \frac{1}{2}(A - A^T)\)

Where \(A^T\) denotes the transpose of matrix \(A\).

Matrix Transposition

First, let's find the transpose of the given matrix \(A\):

\[A = \begin{pmatrix}4&2&-3\\\ 1&3&-6\\\ -5&0&-7\end{pmatrix}\]

To find \(A^T\), we interchange the rows and columns of matrix \(A\):

\[A^T = \begin{pmatrix}4&1&-5\\\ 2&3&0\\\ -3&-6&-7\end{pmatrix}\]

Calculating the Symmetric Matrix P

Next, we calculate the symmetric part \(P\) using the formula \(P = \frac{1}{2}(A + A^T)\).

First, find the sum \(A + A^T\):

\[A + A^T = \begin{pmatrix}4&2&-3\\\ 1&3&-6\\\ -5&0&-7\end{pmatrix} + \begin{pmatrix}4&1&-5\\\ 2&3&0\\\ -3&-6&-7\end{pmatrix}\]

\[A + A^T = \begin{pmatrix}4+4&2+1&-3+(-5)\\\ 1+2&3+3&-6+0\\\ -5+(-3)&0+(-6)&-7+(-7)\end{pmatrix}\]

\[A + A^T = \begin{pmatrix}8&3&-8\\\ 3&6&-6\\\ -8&-6&-14\end{pmatrix}\]

Now, multiply by \(\frac{1}{2}\) to get \(P\):

\[P = \frac{1}{2}\begin{pmatrix}8&3&-8\\\ 3&6&-6\\\ -8&-6&-14\end{pmatrix} = \begin{pmatrix}4&1.5&-4\\\ 1.5&3&-3\\\ -4&-3&-7\end{pmatrix}\]

To verify that \(P\) is a symmetric matrix, check if \(P = P^T\). Interchanging rows and columns of \(P\) yields the same matrix, confirming its symmetric nature.

Calculating the Skew-Symmetric Matrix Q

Now, we calculate the skew-symmetric part \(Q\) using the formula \(Q = \frac{1}{2}(A - A^T)\).

First, find the difference \(A - A^T\):

\[A - A^T = \begin{pmatrix}4&2&-3\\\ 1&3&-6\\\ -5&0&-7\end{pmatrix} - \begin{pmatrix}4&1&-5\\\ 2&3&0\\\ -3&-6&-7\end{pmatrix}\]

\[A - A^T = \begin{pmatrix}4-4&2-1&-3-(-5)\\\ 1-2&3-3&-6-0\\\ -5-(-3)&0-(-6)&-7-(-7)\end{pmatrix}\]

\[A - A^T = \begin{pmatrix}0&1&2\\\ -1&0&-6\\\ -2&6&0\end{pmatrix}\]

Now, multiply by \(\frac{1}{2}\) to get \(Q\):

\[Q = \frac{1}{2}\begin{pmatrix}0&1&2\\\ -1&0&-6\\\ -2&6&0\end{pmatrix} = \begin{pmatrix}0&0.5&1\\\ -0.5&0&-3\\\ -1&3&0\end{pmatrix}\]

To verify that \(Q\) is a skew-symmetric matrix, check if \(Q = -Q^T\). The diagonal elements are zero, and the elements off the main diagonal are negative of their transposes, confirming its skew-symmetric nature.

Summing the Symmetric and Skew-Symmetric Matrices

Finally, we express the original matrix \(A\) as the sum of \(P\) and \(Q\):

\[A = P + Q = \begin{pmatrix}4&1.5&-4\\\ 1.5&3&-3\\\ -4&-3&-7\end{pmatrix} + \begin{pmatrix}0&0.5&1\\\ -0.5&0&-3\\\ -1&3&0\end{pmatrix}\]

Let's add the corresponding elements:

\[P + Q = \begin{pmatrix}4+0&1.5+0.5&-4+1\\\ 1.5+(-0.5)&3+0&-3+(-3)\\\ -4+(-1)&-3+3&-7+0\end{pmatrix}\]

\[P + Q = \begin{pmatrix}4&2&-3\\\ 1&3&-6\\\ -5&0&-7\end{pmatrix}\]

This result matches the original matrix \(A\), confirming the correct decomposition.

Comparing with Options

Comparing our derived symmetric matrix \(P\) and skew-symmetric matrix \(Q\) with the given options, we find that the decomposition matches option 4.

Thus, the correct expression for matrix \(A\) as the sum of a symmetric and a skew-symmetric matrix is:

\[\begin{pmatrix}4&1.5&-4\\\ 1.5&3&-3\\\ -4&-3&-7\end{pmatrix} + \begin{pmatrix}0&0.5&1\\\ -0.5&0&-3\\\ -1&3&0\end{pmatrix}\]