Question

If \[ A = \begin{pmatrix} 2 & 5 \\ 0 & 1 \\ 3 & 0 \end{pmatrix}, \quad A^{-1} = \begin{pmatrix} 3 & -1 \\ -6 & 6 \\ -5 & 2 \end{pmatrix} \] then the values of \( \alpha \) and \( \beta \) are, respectively

• A. 15, 5
• B. -15, 5
• C. 15, -5
• D. -15, -5

Hint:

To find values from a matrix and its inverse, always check that their product is the identity matrix and use matrix multiplication to find unknowns.

Answer 1

The Correct Option is B

Step 1: Using the properties of the inverse matrix.
We are given the matrix \( A \) and its inverse \( A^{-1} \). The product of \( A \) and \( A^{-1} \) should be the identity matrix: \[ A \cdot A^{-1} = I \] We calculate the product of \( A \) and \( A^{-1} \) to find the values of \( \alpha \) and \( \beta \).

Step 2: Multiplying the matrices.
Multiplying \( A \) and \( A^{-1} \), we obtain: \[ \begin{pmatrix} 2 & 5 \\ 0 & 1 \\ 3 & 0 \end{pmatrix} \cdot \begin{pmatrix} 3 & -1 \\ -6 & 6 \\ -5 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{pmatrix} \] The resulting matrix matches the identity matrix, and we determine that the values of \( \alpha \) and \( \beta \) are \( -15 \) and \( 5 \), respectively.

Step 3: Conclusion.
Thus, the values of \( \alpha \) and \( \beta \) are \( -15 \) and \( 5 \), which makes option (B) the correct answer.