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Problem 1

# Show that the matrices are inverses of each other by showing that their product is the identity matrix $$I$$. $$\left[\begin{array}{ll}1 & -3 \\ 1 & -2\end{array}\right]$$ and $$\left[\begin{array}{ll}-2 & 3 \\ -1 & 1\end{array}\right]$$

Expert verified
We will multiply the given matrices $$\textbf{A}$$ and $$\textbf{B}$$ and check if the result is an identity matrix: $$\textbf{A} \cdot \textbf{B} = \left[\begin{array}{ll}1 & -3 \\\ 1 & -2\end{array}\right] \cdot \left[\begin{array}{ll}-2 & 3 \\\ -1 & 1\end{array}\right]$$ Calculating the elements of the resulting matrix $$\textbf{C}$$, we get: $$\textbf{C} = \left[\begin{array}{ll}1 & 0 \\\ 0 & 1\end{array}\right]$$ Since the product of $$\textbf{A}$$ and $$\textbf{B}$$ results in an identity matrix, the matrices $$\textbf{A}$$ and $$\textbf{B}$$ are inverses of each other.
See the step by step solution

## Step 1: Multiply the two matrices together

First, let's multiply the given matrices $$\textbf{A}$$ and $$\textbf{B}$$ together: $$\textbf{A} = \left[\begin{array}{ll}1 & -3 \\\ 1 & -2\end{array}\right]$$, $$\textbf{B} = \left[\begin{array}{ll}-2 & 3 \\\ -1 & 1\end{array}\right]$$ $$\textbf{C} = \textbf{A} \cdot \textbf{B} = \left[\begin{array}{ll}1 & -3 \\\ 1 & -2\end{array}\right] \cdot \left[\begin{array}{ll}-2 & 3 \\\ -1 & 1\end{array}\right]$$

## Step 2: Calculate the elements of the resulting matrix

To find the elements of the resulting matrix $$\textbf{C}$$, let's use matrix multiplication rules. Multiply the elements of each row in $$\textbf{A}$$ by the corresponding elements of each column in $$\textbf{B}$$, and then add the results: $$c_{11} = (1 \times -2) + (-3 \times -1) = -2 + 3 = 1$$ $$c_{12} = (1 \times 3) + (-3 \times 1) = 3 - 3 = 0$$ $$c_{21} = (1 \times -2) + (-2 \times -1) = -2 + 2 = 0$$ $$c_{22} = (1 \times 3) + (-2 \times 1) = 3 - 2 = 1$$

## Step 3: Write the resulting matrix

Now that we have calculated all the elements of matrix $$\textbf{C}$$, let's write them as a matrix: $$\textbf{C} = \left[\begin{array}{ll}1 & 0 \\\ 0 & 1\end{array}\right]$$

## Step 4: Compare the matrices

Since the product of $$\textbf{A}$$ and $$\textbf{B}$$ resulted in an identity matrix, we can conclude that the matrices $$\textbf{A}$$ and $$\textbf{B}$$ are inverses of each other.

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