Question

Let α and β be real numbers. Consider a \(3 \times 3\) matrix A such that \(A^2 = 3A + \alpha I\). If \(A^4 = 21A + \beta I\), then

• A. α = 1
• B. α=4
• C. β=8
• D. β=-8

Answer 1

The Correct Option is D

Step 1: Start with the given equation 

We are given two equations that describe the matrix \( A \):

\( A^2 = 3A + \alpha I \quad \cdots (1)\)

\( A^4 = 21A + \beta I \quad \cdots (2)\)

Where \( A \) is a matrix, and \( I \) is the identity matrix.

Step 2: Express \( A^4 \) in terms of \( A^2 \)

We need to express \( A^4 \) in terms of \( A^2 \). To begin, we square equation (1) to get the expression for \( A^2 \):

\( A^2 = 3A + \alpha I \)

Now, multiply both sides of this equation by \( A \) to find the expression for \( A^3 \):

\( A^3 = A \cdot A^2 = A(3A + \alpha I) = 3A^2 + \alpha A\)

Now, substitute \( A^2 = 3A + \alpha I \) into this equation to simplify:

\( A^3 = 3(3A + \alpha I) + \alpha A = 9A + 3\alpha I + \alpha A\)

Now, expand to get the expression for \( A^4 \):

\( A^4 = (9 + \alpha)A^2 + 3\alpha A\)

Now, substitute the expression for \( A^2 \) from equation (1) into this new expression for \( A^4 \):

\( A^4 = (9 + \alpha)(3A + \alpha I) + 3\alpha A\)

Now, expand this equation:

\( A^4 = A(27 + 6\alpha) + \alpha(9 + \alpha)I\)

Step 3: Simplify the expression

From the equation for \( A^4 \), we now have:

\( A^4 = A(27 + 6\alpha) + \alpha(9 + \alpha)I\)

To make this consistent with equation (2), compare the coefficients of \( A \) and \( I \):

For the coefficient of \( A \), we have:

\( 27 + 6\alpha = 21 \Rightarrow \alpha = -1\)

For the coefficient of \( I \), we have:

\( \beta = \alpha(9 + \alpha) = -8\)

Step 4: Conclusion

Thus, the values of \( \alpha \) and \( \beta \) are:

\( \alpha = -1, \quad \beta = -8\)

Therefore, the correct values for \( \alpha \) and \( \beta \) are \( \alpha = -1 \) and \( \beta = -8 \), as derived from the matrix equations.