The trace of matrix $\mathbf{A} = \mathbf{B}^T + 2\mathbf{C} - 3\mathbf{I}_3$, where $\mathbf{B}=\begin{pmatrix}1&2&1\\-1&1&-1\\-1&-1&1\\\end{pmatrix}$,  $\mathbf{C}=\begin{pmatrix}2&3&1 \\-2&1&3\\-3&-1&1\\\end{pmatrix}$ and $I_3$ being the 3×3 identity matrix equals :

To find the trace of matrix $\mathbf{A} = \mathbf{B}^T + 2\mathbf{C} - 3\mathbf{I}_3$, we start by computing each term individually and then combining them.

First, calculate the transpose of $\mathbf{B}$:
\[
\mathbf{B}^T = \begin{pmatrix}
1 & -1 & -1 \\
2 & 1 & -1 \\
1 & -1 & 1 \\
\end{pmatrix}
\]

Next, multiply matrix $\mathbf{C}$ by 2:
\[
2\mathbf{C} = 2 \begin{pmatrix}
2 & 3 & 1 \\
-2 & 1 & 3 \\
-3 & -1 & 1 \\
\end{pmatrix} = \begin{pmatrix}
4 & 6 & 2 \\
-4 & 2 & 6 \\
-6 & -2 & 2 \\
\end{pmatrix}
\]

Now, the 3×3 identity matrix $\mathbf{I}_3$ is:
\[
\mathbf{I}_3 = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{pmatrix}
\]

Multiplying this by 3:
\[
3\mathbf{I}_3 = 3 \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{pmatrix} = \begin{pmatrix}
3 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 3 \\
\end{pmatrix}
\]

Add the matrices $\mathbf{B}^T$, $2\mathbf{C}$, and $-3\mathbf{I}_3$ together to get $\mathbf{A}$:
\[
\mathbf{A} = \mathbf{B}^T + 2\mathbf{C} - 3\mathbf{I}_3 = \begin{pmatrix}
1 & -1 & -1 \\
2 & 1 & -1 \\
1 & -1 & 1 \\
\end{pmatrix} + \begin{pmatrix}
4 & 6 & 2 \\
-4 & 2 & 6 \\
-6 & -2 & 2 \\
\end{pmatrix} - \begin{pmatrix}
3 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 3 \\
\end{pmatrix}
\]

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

The trace of a matrix is the sum of its diagonal elements. Therefore, for matrix $\mathbf{A}$:
\[
\text{trace}(\mathbf{A}) = 2 + 0 + 0 = 2
\]

Thus, the trace of matrix $\mathbf{A}$ is $2$.

Would you like more details or have any questions?

Here are five questions you might consider next:
1. What is the determinant of a given 3x3 matrix?
2. How do you find the eigenvalues of a 3x3 matrix?
3. What is the inverse of a given 2x2 or 3x3 matrix?
4. How do you perform matrix multiplication for two 3x3 matrices?
5. How do you solve a system of linear equations using matrices?

**Tip:** When working with matrices, always check for properties like symmetry and orthogonality, as they can simplify many operations.

Learn how to calculate the trace of matrix A using matrix B transpose, matrix C multiplied by 2, and the 3x3 identity matrix multiplied by -3. Understand matrix operations and scalar multiplication in detail. Suitable for advanced high school and undergraduate students.

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