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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 \(\left[\begin{array}{ll}-2 & 3 \\ -1 & 1\end{array}\right]\)

Short Answer

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.
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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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