Let A=(α−16−1β), α>0, such that det(A)=0 and α+β=1. If I denotes the 2×2 identity matrix, then the matrix (I+A)8 is:
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Step-by-Step Solution
Step 1: Use given conditions to find α and β
We are given two conditions: the determinant of matrix A is zero, and the sum of α and β is 1. We will use these conditions to find the values of α and β. First, we calculate the determinant of A and set it to zero. Then, we substitute β=1−α into the determinant equation.
Step 2: Solve for α
Substitute β=1−α into the determinant equation. This gives us a quadratic equation in terms of α. We simplify and rearrange the terms to get a standard quadratic form.
Step 3: Re-evaluate the determinant calculation
Let's re-evaluate the determinant calculation. The determinant of a 2×2 matrix (acbd) is ad-bc. So, det(A)=(α−1)β−(−1)(6)=(α−1)β+6. Substituting β=1−α into this equation gives us (α−1)(1−α)+6=0. Expanding this, we get α−α2−1+α+6=0, which simplifies to −α2+2α+5=0, or α2−2α−5=0. This is the correct quadratic equation for α. However, the problem statement implies a simpler solution for A or I+A. Let's check if there's a simpler way to use the given conditions.
Step 4: Re-examine the matrix A and conditions
The problem seems to lead to a quadratic equation for α with irrational roots, which would make (I+A)8 very complex to calculate directly. This suggests there might be a property of the matrix A or I+A that simplifies the calculation. Let's re-examine the given conditions and the structure of matrix A.
Step 5: Consider the characteristic equation of A
The characteristic equation of a matrix A is given by det(A−λI)=0. Expanding this for matrix A, we get a quadratic equation in λ. This equation relates to the eigenvalues of A.
Step 6: Simplify the characteristic equation using given conditions
From the characteristic equation, we know that the sum of the diagonal elements (trace) is Tr(A)=(α−1)+β=(α+β)−1. The determinant is det(A)=(α−1)β+6. Using the given conditions α+β=1 and det(A)=0, we substitute these into the characteristic equation. This simplifies to λ2−(1−1)λ+0=0, which means λ2=0.
Step 7: Determine the nature of matrix A and calculate (I+A)^8
Since the characteristic equation is λ2=0, both eigenvalues of A are 0. This means A is a nilpotent matrix of index 2, i.e., A2=0. We can use the binomial expansion for (I+A)8. Since A2=0, all higher powers of A (A3,A4,…) will also be zero. Therefore, the expansion simplifies significantly to I + 8A.