We are tasked to solve the matrix equation:

$[I]BC = C^{-1}B$

where:

• $C^{-1} = \{(1,2), (3,4)\}$,
• $B = \{(5,6), (7,8)\}$,
• $[I]$ is the identity matrix.

Step 1: Represent $C^{-1}$ and $B$ as matrices

$C^{-1} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}.$

Step 2: Compute $C^{-1}B$

Multiply $C^{-1}$ and $B$: $C^{-1}B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}.$

Perform the multiplication:

= \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}.$$ ### Step 3: Represent $$[I]BC$$ Since $$[I]$$ is the identity matrix, $$[I]BC = BC$$. Therefore: $$BC = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}.$$ Compute $$BC$$: $$BC = \begin{bmatrix} (5)(1) + (6)(3) & (5)(2) + (6)(4) \\ (7)(1) + (8)(3) & (7)(2) + (8)(4) \end{bmatrix} = \begin{bmatrix} 23 & 34 \\ 31 & 46 \end{bmatrix}.$$ ### Step 4: Compare $$[I]BC$$ and $$C^{-1}B$$ From the calculations: - $$[I]BC = BC = \begin{bmatrix} 23 & 34 \\ 31 & 46 \end{bmatrix}$$, - $$C^{-1}B = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$$. Clearly, $$[I]BC \neq C^{-1}B$$. Therefore, the equation is **not satisfied**. --- ### Would you like additional details or have any questions? Here are related questions to explore: 1. How do we verify if a given matrix is invertible? 2. What properties do matrix products like $$C^{-1}B$$ satisfy? 3. Can you explain how the identity matrix interacts with other matrices? 4. Why is matrix multiplication not commutative, and how does it affect this problem? 5. How do eigenvalues relate to matrix inverses? **Tip:** Always check dimensions before performing matrix multiplication to ensure compatibility!