If A = begin{bmatrix} 0 & 1 1 & 0 end{bmatri | vedclass

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(A) Given $A = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}$.First, calculate $A^2 = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 &

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$8A$

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(A) Given $A = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}$.First, calculate $A^2 = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = I$.Since $A^2 = I$, it follows that $A^3 = A^2 \cdot A = I \cdot A = A$.Now, expand $(A+I)^3 = A^3 + 3A^2I + 3AI^2 + I^3 = A + 3I + 3A + I = 4A + 4I$.Next, expand $(A-I)^3 = A^3 - 3A^2I + 3AI^2 - I^3 = A - 3I + 3A - I = 4A - 4I$.Finally, add the two expressions: $(4A+4I) + (4A-4I) = 8A$.

If $A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, then $(A+I)^3 + (A-I)^3 = \dots$

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