Given the matrix $A$ as follows:

$A = \begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & 0 & -a \end{bmatrix}$

We are tasked with finding $A^3 - A(a^2 + b^2 + c^2)$.

Step 1: Matrix Multiplication - $A^2$

To compute $A^3$, we first need $A^2$.

$A^2 = A \cdot A$ $A^2 = \begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & 0 & -a \end{bmatrix} \cdot \begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & 0 & -a \end{bmatrix}$ Performing matrix multiplication:

$A^2 = \begin{bmatrix} -(b^2 + c^2) & 0 & 0 \\ 0 & -(a^2 + c^2) & 0 \\ 0 & 0 & -(a^2 + b^2) \end{bmatrix}$

Step 2: Matrix Multiplication - $A^3$

Now, compute $A^3 = A \cdot A^2$:

$A^3 = \begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & 0 & -a \end{bmatrix} \cdot \begin{bmatrix} -(b^2 + c^2) & 0 & 0 \\ 0 & -(a^2 + c^2) & 0 \\ 0 & 0 & -(a^2 + b^2) \end{bmatrix}$ Performing matrix multiplication:

$A^3 = \begin{bmatrix} 0 & -(a^2 + c^2)c & (a^2 + b^2)b \\ (a^2 + c^2)c & 0 & -(a^2 + b^2)a \\ -(a^2 + b^2)b & (a^2 + b^2)a & 0 \end{bmatrix}$

Step 3: Compute $A(a^2 + b^2 + c^2)$

Next, calculate $A(a^2 + b^2 + c^2)$, where $A$ is multiplied by the scalar $a^2 + b^2 + c^2$:

$A(a^2 + b^2 + c^2) = (a^2 + b^2 + c^2) \cdot \begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & 0 & -a \end{bmatrix}$ This yields:

$A(a^2 + b^2 + c^2) = \begin{bmatrix} 0 & (a^2 + b^2 + c^2)c & -(a^2 + b^2 + c^2)b \\ -(a^2 + b^2 + c^2)c & 0 & (a^2 + b^2 + c^2)a \\ (a^2 + b^2 + c^2)b & -(a^2 + b^2 + c^2)a & 0 \end{bmatrix}$

Step 4: Compute $A^3 - A(a^2 + b^2 + c^2)$

Finally, subtract $A(a^2 + b^2 + c^2)$ from $A^3$:

$A^3 - A(a^2 + b^2 + c^2) = \begin{bmatrix} 0 & -(a^2 + c^2)c & (a^2 + b^2)b \\ (a^2 + c^2)c & 0 & -(a^2 + b^2)a \\ -(a^2 + b^2)b & (a^2 + b^2)a & 0 \end{bmatrix} - \begin{bmatrix} 0 & (a^2 + b^2 + c^2)c & -(a^2 + b^2 + c^2)b \\ -(a^2 + b^2 + c^2)c & 0 & (a^2 + b^2 + c^2)a \\ (a^2 + b^2 + c^2)b & -(a^2 + b^2 + c^2)a & 0 \end{bmatrix}$

This simplifies to:

$A^3 - A(a^2 + b^2 + c^2) = \begin{bmatrix} 0 & -2c^3 & 2b^3 \\ 2c^3 & 0 & -2a^3 \\ -2b^3 & 2a^3 & 0 \end{bmatrix}$

Thus, the final result is:

$A^3 - A(a^2 + b^2 + c^2) = \begin{bmatrix} 0 & -2c^3 & 2b^3 \\ 2c^3 & 0 & -2a^3 \\ -2b^3 & 2a^3 & 0 \end{bmatrix}$

Let me know if you want more details or have any questions!

Related Questions:

1. What is the geometric interpretation of a skew-symmetric matrix like $A$?
2. How do the eigenvalues of skew-symmetric matrices behave?
3. Can you explain the properties of cubic powers of matrices?
4. How does the Cayley-Hamilton theorem apply to skew-symmetric matrices?
5. What are the applications of matrices like $A$ in physics?

Tip:

For skew-symmetric matrices like $A$, all the eigenvalues are purely imaginary or zero.