
3)  How do you know if two matrices are inverses of one another?


Two matrices \( A \) and \( B \) are inverses of one another if and only if their product (in both orders) is the identity matrix. Specifically, the following conditions must hold:

1. \( A \times B = I \)
2. \( B \times A = I \)

Here, \( I \) is the identity matrix of the same dimension as \( A \) and \( B \). The identity matrix is a square matrix with 1s on the diagonal and 0s everywhere else.

For example, for two \( 2 \times 2 \) matrices \( A \) and \( B \), they are inverses if:

\[
A \times B = B \times A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\]

If both conditions are satisfied, then \( A \) and \( B \) are inverses of each other.

Would you like to see an example calculation of inverse matrices?

Here are 5 related questions:
1. How do you calculate the inverse of a \( 2 \times 2 \) matrix?
2. What are the conditions for a matrix to have an inverse?
3. Can non-square matrices have inverses? Why or why not?
4. What is the inverse of a matrix in terms of its determinant?
5. How do you check if a matrix is invertible?

**Tip:** Not all matrices have inverses. A matrix must be square (same number of rows and columns) and have a non-zero determinant to be invertible.

Explore how to determine if two matrices are inverses of each other. Learn about the conditions that must be satisfied, including the product of matrices resulting in the identity matrix. Suitable for advanced high school and college-level students.

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