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(B) Let $A = \begin{bmatrix} 1 & 2 & 3 \ 0 & 1 & 2 \ 0 & 0 & 1 \end{bmatrix}$.First,we calculate the determinant $|A| = 1(1 	imes 1 - 2 	imes 0) -

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$\begin{bmatrix} 1 & -2 & 1 \ 0 & 1 & -2 \ 0 & 0 & 1 \end{bmatrix}$

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(B) Let $A = \begin{bmatrix} 1 & 2 & 3 \ 0 & 1 & 2 \ 0 & 0 & 1 \end{bmatrix}$.First,we calculate the determinant $|A| = 1(1 	imes 1 - 2 	imes 0) - 2(0 	imes 1 - 2 	imes 0) + 3(0 	imes 0 - 1 	imes 0) = 1(1) = 1$.Since $|A| 
eq 0$,the inverse $A^{-1}$ exists.Next,we find the matrix of cofactors $C_{ij}$:$C_{11} = +|\begin{smallmatrix} 1 & 2 \ 0 & 1 \end{smallmatrix}| = 1, C_{12} = -|\begin{smallmatrix} 0 & 2 \ 0 & 1 \end{smallmatrix}| = 0, C_{13} = +|\begin{smallmatrix} 0 & 1 \ 0 & 0 \end{smallmatrix}| = 0$$C_{21} = -|\begin{smallmatrix} 2 & 3 \ 0 & 1 \end{smallmatrix}| = -2, C_{22} = +|\begin{smallmatrix} 1 & 3 \ 0 & 1 \end{smallmatrix}| = 1, C_{23} = -|\begin{smallmatrix} 1 & 2 \ 0 & 0 \end{smallmatrix}| = 0$$C_{31} = +|\begin{smallmatrix} 2 & 3 \ 1 & 2 \end{smallmatrix}| = 4-3 = 1, C_{32} = -|\begin{smallmatrix} 1 & 3 \ 0 & 2 \end{smallmatrix}| = -2, C_{33} = +|\begin{smallmatrix} 1 & 2 \ 0 & 1 \end{smallmatrix}| = 1$Thus,$Adj(A) = \begin{bmatrix} 1 & -2 & 1 \ 0 & 1 & -2 \ 0 & 0 & 1 \end{bmatrix}$.Finally,$A^{-1} = \frac{1}{|A|} Adj(A) = \frac{1}{1} \begin{bmatrix} 1 & -2 & 1 \ 0 & 1 & -2 \ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & -2 & 1 \ 0 & 1 & -2 \ 0 & 0 & 1 \end{bmatrix}$.

The inverse of $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix}$ is

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