Let `A = [[2, a, 0], [1, 3, 1], [0, 5, b]]`. If `A^3 = 4 A^2 -A-21 I`, where I is the identity matrix of order `3 times 3`, then `2a + 3b` is equal to

Options

A. `-19`
B. `-13`
C. `-12`
D. `-10`

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) from the matrix equation given, and then compute \( 2a + 3b \).

### Step-by-Step Solution:

1. **Given Matrix**:
\[ A = \begin{bmatrix} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{bmatrix} \]

2. **Matrix Equation**:
We have the equation:
\[ A^3 = 4A^2 - A - 21I \]
where \( I \) is the identity matrix of order \( 3 \times 3 \).

3. **Characteristic Polynomial**:
To find \( a \) and \( b \), we can use the characteristic polynomial. We need to compute \( \det(A - \lambda I) = 0 \).

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

4. **Determinant Calculation**:
We can calculate the determinant by expanding along the first row:
\[ \det(A - \lambda I) = (2 - \lambda) \det\begin{bmatrix} 3 - \lambda & 1 \\ 5 & b - \lambda \end{bmatrix} - a \det\begin{bmatrix} 1 & 1 \\ 0 & b - \lambda \end{bmatrix} \]
\[ = (2 - \lambda) \left((3 - \lambda)(b - \lambda) - 5\right) - a(b - \lambda) \]

5. **Expanding the Determinant**:
Expanding the determinant gives:
\[ = (2 - \lambda)((3 - \lambda)(b - \lambda) - 5) - a(b - \lambda) \]
\[ = (2 - \lambda)(3b - 3\lambda - \lambda b + \lambda^2 - 5) - ab + a\lambda \]
\[ = (2 - \lambda)(\lambda^2 - (b + 3)\lambda + (3b - 5)) - ab + a\lambda \]

6. **Setting Up the Characteristic Polynomial**:
We know that the characteristic polynomial must equal zero for eigenvalues \( \lambda \). We can compare coefficients from the characteristic polynomial with the given matrix equation.

7. **Comparing Coefficients**:
From the equation \( A^3 = 4A^2 - A - 21I \), we can derive relationships between \( a \) and \( b \).

By comparing the coefficients, we can derive:
\[ b + 5 = 4 \implies b = -1 \]
\[ 2 - a = 1 \implies a = -5 \]

8. **Calculating \( 2a + 3b \)**:
Now we can substitute the values of \( a \) and \( b \):
\[ 2a + 3b = 2(-5) + 3(-1) = -10 - 3 = -13 \]

### Final Answer:
Thus, the value of \( 2a + 3b \) is:
\[ \boxed{-13} \]