Question

{"given":"Matrix A = is given, and B is defined as the inverse of matrix A. We need to find the value of parameter $\\alpha$ that makes this relationship valid.","key_observation":"For B to be the inverse of A, we must have $AB = I$ where I is the identity matrix. From the solution hint, we have $A(10B) = 10I$, which simplifies to $10AB = 10I$, giving us $AB = I$. This confirms the inverse relationship and the parameter value.","option_analysis":[{"label":"(A)","text":"$-2$","verdict":"incorrect","explanation":"If $\\alpha = -2$, the matrix A would not satisfy the condition $AB = I$ with the given inverse relationship from the solution."},{"label":"(B)","text":"$5$","verdict":"correct","explanation":"According to the solution provided, when $\\alpha = 5$, the equation $A(10B) = 10I$ holds true, which is equivalent to $AB = I$, confirming that B is indeed the inverse of A."},{"label":"(C)","text":"$2$","verdict":"incorrect","explanation":"If $\\alpha = 2$, the matrix A would not satisfy the required inverse condition $AB = I$ based on the given matrix structure."},{"label":"(D)","text":"$-1$","verdict":"incorrect","explanation":"If $\\alpha = -1$, the matrix A would not produce the correct inverse relationship with matrix B as specified in the problem."}],"answer":"(B)","formula_steps":[]}