If `A=diag(2,-1,3), B=diag(-1,3,2)` then `A^(2)B`

Options

A. diag(5,4,11)
B. diag`(-4,3,18)`
C. diag(3,1,8)
D. diag(3,1,19)

Answer

To solve the problem, we need to find \( A^2 B \) where \( A \) and \( B \) are diagonal matrices.

Given:
\[ A = \text{diag}(2, -1, 3) \quad \text{and} \quad B = \text{diag}(-1, 3, 2) \]

### Step 1: Calculate \( A^2 \)

Since \( A \) is a diagonal matrix, squaring it is straightforward. The square of a diagonal matrix is obtained by squaring each of its diagonal elements.

\[ A^2 = \text{diag}(2^2, (-1)^2, 3^2) = \text{diag}(4, 1, 9) \]

### Step 2: Calculate \( A^2 B \)

Now, we need to multiply \( A^2 \) with \( B \). Again, since both \( A^2 \) and \( B \) are diagonal matrices, the product \( A^2 B \) is also a diagonal matrix where each diagonal entry is the product of the corresponding diagonal entries of \( A^2 \) and \( B \).

\[ A^2 B = \text{diag}(4, 1, 9) \cdot \text{diag}(-1, 3, 2) = \text{diag}(4 \cdot (-1), 1 \cdot 3, 9 \cdot 2) \]

Calculating each entry:
- First diagonal entry: \( 4 \cdot (-1) = -4 \)
- Second diagonal entry: \( 1 \cdot 3 = 3 \)
- Third diagonal entry: \( 9 \cdot 2 = 18 \)

Thus, we have:
\[ A^2 B = \text{diag}(-4, 3, 18) \]

### Final Result

The final result is:
\[ A^2 B = \text{diag}(-4, 3, 18) \]

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