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Let A = bmatrix -1 & 1 & -1 1 & 0 & 1 0 & 0 & 1 bmatrix satisfy A^2 + (adj(adj(A))) + (adj(A)(adj(adj(A)))) = bmatrix 2 & -2 & 2 -2 & 0 & -1 0 & 0 & -1 bmatrix for some , R . Then ( - )^2 is equal to _______

Correct answer

0

Step-by-step solution

The determinant of matrix A is: |A| = -1(0 - 0) - 1(1 - 0) - 1(0 - 0) = -1 Using the properties of the adjoint of a 3 3 matrix: adj(adj(A)) = |A|³⁻²A = -A adj(A)(adj(adj(A))) = adj(A)(-A) = -|A|I = -(-1)I = I The given equation simplifies to: A^2 - A + I = bmatrix 2 & -2 & 2 -2 & 0 & -1 0 & 0 & -1 bmatrix Calculating A^2 : A^2 = bmatrix -1 & 1 & -1 1 & 0 & 1 0 & 0 & 1 bmatrix bmatrix -1 & 1 & -1 1 & 0 & 1 0 & 0 & 1 bmatrix = bmatrix 2 & -1 & 1 -1 & 1 & 0 0 & 0 & 1 bmatrix Substituting A^2 , A , and I into the simpl

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