To solve the inverse of matrix $A^{-1}$ for both matrices given in the image, we will use the Gauss-Jordan elimination method, which involves transforming the augmented matrix $[A|I]$ into $[I|A^{-1}]$.

a) For matrix $A = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix}$:

1. Step 1: Write the augmented matrix $[A | I]$, where $I$ is the identity matrix:

   1 & 1 & 1 & | & 1 & 0 & 0 \\ 0 & 1 & 1 & | & 0 & 1 & 0 \\ 0 & 0 & 1 & | & 0 & 0 & 1 \\ \end{pmatrix}$$

2. Step 2: Apply row operations to turn the left-hand side into the identity matrix.

   • Row 1 is already fine. Focus on Row 2.
   • Subtract Row 2 from Row 1: $R1' = R1 - R2 \implies \begin{pmatrix} 1 & 0 & 0 & | & 1 & -1 & 1 \end{pmatrix}$

   Thus, the final matrix becomes:

   1 & 0 & 0 & | & 1 & -1 & 1 \\ 0 & 1 & 0 & | & 0 & 1 & -1 \\ 0 & 0 & 1 & | & 0 & 0 & 1 \end{pmatrix}$$

3. Conclusion: The inverse of matrix $A$ is:

   1 & -1 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{pmatrix}$$

b) For matrix $A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & 3 & 1 \\ 3 & 4 & 0 \end{pmatrix}$:

1. Step 1: Write the augmented matrix $[A | I]$:

   1 & 1 & 1 & | & 1 & 0 & 0 \\ 2 & 3 & 1 & | & 0 & 1 & 0 \\ 3 & 4 & 0 & | & 0 & 0 & 1 \end{pmatrix}$$

2. Step 2: Apply row operations to transform $A$ into the identity matrix.

   • First, subtract 2 times Row 1 from Row 2: $R2' = R2 - 2R1 = \begin{pmatrix} 0 & 1 & -1 & | & -2 & 1 & 0 \end{pmatrix}$

   • Subtract 3 times Row 1 from Row 3: $R3' = R3 - 3R1 = \begin{pmatrix} 0 & 1 & -3 & | & -3 & 0 & 1 \end{pmatrix}$

   • Now subtract Row 2 from Row 3 to get: $R3'' = R3' - R2' = \begin{pmatrix} 0 & 0 & -2 & | & -1 & -1 & 1 \end{pmatrix}$

   • Divide Row 3 by -2: $R3''' = \begin{pmatrix} 0 & 0 & 1 & | & \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \end{pmatrix}$

   • Add 1 times Row 3 to Row 2: $R2'' = \begin{pmatrix} 0 & 1 & 0 & | & -\frac{3}{2} & \frac{1}{2} & -\frac{1}{2} \end{pmatrix}$

   • Subtract 1 times Row 3 from Row 1: $R1'' = \begin{pmatrix} 1 & 0 & 0 & | & \frac{5}{2} & -\frac{1}{2} & \frac{1}{2} \end{pmatrix}$

3. Conclusion: The inverse of matrix $A$ is:

   \frac{5}{2} & -\frac{1}{2} & \frac{1}{2} \\ -\frac{3}{2} & \frac{1}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} \end{pmatrix}$$

Let me know if you'd like any details about the steps or if you have further questions!

Related questions:

1. How is the determinant of a matrix related to its invertibility?
2. Can a matrix with a determinant of 0 have an inverse? Why or why not?
3. What is the significance of row reduction in finding the inverse of a matrix?
4. How does the Gauss-Jordan elimination method differ from other methods of finding the inverse?
5. Can non-square matrices have inverses? Why or why not?

Tip: Always check if a matrix is invertible by calculating its determinant before attempting to find its inverse!