Let A be the 3x3 matrix with entries: A[1][1] = lambda² - 3*lambda + 2, A[1][2] = 3, A[1][3] = -6; A[2][1] = -3, A[2][2] = lambda³ - 6*lambda² + 11*lambda - 6, A[2][3] = 4; A[3][1] = 6, A[3][2] = -4, A[3][3] = tan(pi/4) - 1. If A is skew-symmetric, find all possible values of lambda.

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Accepted Answer

1. Skew-symmetric requires diagonal = 0. A[3][3]=tan(pi/4)-1=1-1=0 (always). A[1][1]=lambda²-3lambda+2=0 => (lambda-1)(lambda-2)=0 => lambda=1 or 2. A[2][2]=lambda³-6lambda²+11lambda-6=0 => (lambda-1)(lambda-2)(lambda-3)=0 => lambda=1,2,3. Common values: lambda=1 or 2. Off-diagonal conditions A[1][2]=3, A[2][1]=-3 (consistent: -A[2][1]=3=A[1][2]); A[1][3]=-6, A[3][1]=6 (consistent); A[2][3]=4, A[3][2]=-4 (consistent). Both lambda=1 and lambda=2 work. The answer options include both; selecting lambda=1 as the primary answer (and lambda=2 is equally valid).

Let A be the 3x3 matrix with entries: A[1][1] = lambda² - 3lambda + 2, A[1][2] = 3, A[1][3] = -6; A[2][1] = -3, A[2][2] = lambda³ - 6lambda² + 11*lambda - 6, A[2][3] = 4; A[3][1] = 6, A[3][2] = -4, A[3][3] = tan(pi/4) - 1. If A is skew-symmetric, find all possible values of lambda.

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