Find output of A³-A(a²+b²+c²). A=[[0,c,-b],[-c,0,a],[b,0,-a]

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.

Find the output of A³ - A(a² + b² + c²), where A = [[0, c, -b], [-c, 0, a], [b, 0, -a]]

Explore a detailed solution to the matrix problem A³ - A(a² + b² + c²) for a skew-symmetric matrix A. This problem delves into matrix algebra, focusing on matrix multiplication, properties of skew-symmetric matrices, and advanced linear algebra concepts. Suitable for university students.

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