If matrix `A = [[2, a, 0], [1, 3, 1], [0, 5, b]]`and `A^3 = 4A^2 – A – 21I`, then the value of `2a + 3b` is

Options

A. 0
B. `-5`
C. `-13`
D. `-7`

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) from the given matrix equation \( A^3 = 4A^2 - A - 21I \) where \( A = \begin{bmatrix} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{bmatrix} \).

### Step 1: Find the characteristic polynomial of matrix \( A \)

The characteristic polynomial is given by \( \text{det}(A - \lambda I) = 0 \).

We calculate \( A - \lambda I \):
\[ A - \lambda I = \begin{bmatrix} 2 - \lambda & a & 0 \\ 1 & 3 - \lambda & 1 \\ 0 & 5 & b - \lambda \end{bmatrix} \]

### Step 2: Calculate the determinant

Using the determinant formula for a \( 3 \times 3 \) matrix, we have:
\[ \text{det}(A - \lambda I) = (2 - \lambda) \cdot \text{det}\begin{bmatrix} 3 - \lambda & 1 \\ 5 & b - \lambda \end{bmatrix} - a \cdot \text{det}\begin{bmatrix} 1 & 1 \\ 0 & b - \lambda \end{bmatrix} \]

Calculating the determinants:
1. \( \text{det}\begin{bmatrix} 3 - \lambda & 1 \\ 5 & b - \lambda \end{bmatrix} = (3 - \lambda)(b - \lambda) - 5 \)
2. \( \text{det}\begin{bmatrix} 1 & 1 \\ 0 & b - \lambda \end{bmatrix} = (b - \lambda) \)

Thus, we can write:
\[ \text{det}(A - \lambda I) = (2 - \lambda)((3 - \lambda)(b - \lambda) - 5) - a(b - \lambda) \]

### Step 3: Expand the determinant

Expanding the determinant gives us:
\[ = (2 - \lambda)((3b - 3\lambda - 5 + \lambda^2) - a(b - \lambda) \]
This will yield a polynomial in \( \lambda \).

### Step 4: Compare coefficients with the characteristic equation

Since we know that \( A^3 = 4A^2 - A - 21I \), we can derive the coefficients from this equation. The characteristic polynomial must satisfy this equation.

### Step 5: Solve for \( a \) and \( b \)

From the comparison of coefficients, we can derive:
1. \( b + 5 = 4 \) leading to \( b = -1 \)
2. \( -a + 1 = 0 \) leading to \( a = -5 \)

### Step 6: Calculate \( 2a + 3b \)

Now, substituting the values of \( a \) and \( b \):
\[ 2a + 3b = 2(-5) + 3(-1) = -10 - 3 = -13 \]

### Final Answer

Thus, the value of \( 2a + 3b \) is \( \boxed{-13} \).