Question

Let $A = \begin{bmatrix} 2 & 0 & 1 \\\ 1 & 1 & 0 \\\ 1 & 0 & 1 \end{bmatrix}$, $B = [B_1, B_2, B_3]$, where $B_1, B_2, B_3$ are column matrices, and $AB_1 = \begin{bmatrix} 1 \\\ 0 \\\ 0 \end{bmatrix}$, $AB_2 = \begin{bmatrix} 2 \\\ 3 \\\ 0 \end{bmatrix}$, $AB_3 = \begin{bmatrix} 3 \\\ 2 \\\ 1 \end{bmatrix}$.
If $\alpha = |B|$ and $\beta$ is the sum of all the diagonal elements of B, then $\alpha^3 + \beta^3$ is equal to _____.

Answer 1

Step 1. Define Matrices:
$A = \begin{bmatrix} 2 & 0 & 1 \\\ 1 & 1 & 0 \\\ 1 & 0 & 1 \end{bmatrix}$, $B = [B_1, B_2, B_3]$

where
$B_1 = \begin{bmatrix} x_1 \\\ y_1 \\\ z_1 \end{bmatrix}$, $B_2 = \begin{bmatrix} x_2 \\\ y_2 \\\ z_2 \end{bmatrix}$, $B_3 = \begin{bmatrix} x_3 \\\ y_3 \\\ z_3 \end{bmatrix}$.

Step 2. Equations from Matrix Multiplication:
- For $AB_1 = \begin{bmatrix} 1 \\\ 0 \\\ 0 \end{bmatrix}$, we get:
$\begin{cases} 2x_1 + z_1 = 1 \\\ x_1 + y_1 = 0 \\\ x_1 + z_1 = 0 \end{cases}$

- For $AB_2 = \begin{bmatrix} 2 \\\ 3 \\\ 0 \end{bmatrix}$, we get:
$\begin{cases} 2x_2 + z_2 = 2 \\\ x_2 + y_2 = 3 \\\ x_2 + z_2 = 0 \end{cases}$

- For $AB_3 = \begin{bmatrix} 3 \\\ 2 \\\ 1 \end{bmatrix}$, we get:
$\begin{cases} 2x_3 + z_3 = 3 \\\ x_3 + y_3 = 2 \\\ x_3 + z_3 = 1 \end{cases}$

Step 3. Solving for $B$: Solve these systems of equations to determine the values of $B_1$, $B_2$, and $B_3$.

Step 4. Calculate $\alpha$ and $\beta$:
- $\alpha = |B| = 3 - \beta$ is the sum of the diagonal elements of $B$, which is 1.

Step 5. Find $\alpha^3 + \beta^3$:
$\alpha^3 + \beta^3 = 27 + 1 = 28$