Question

Prove that if there are two matrices \(\mathbf{B}\) and \(\mathbf{C}\) such that \(\mathbf{A B}=\mathbf{I}\) and \(\mathbf{A C}=\mathbf{I},\) then \(\mathbf{B}=\mathbf{C} .\) This shows that a matrix A can have only one inverse.

Short answer

Question: Prove that if there are two matrices B and C such that AB = I and AC = I, then B = C. Answer: We have proven that if AB = I and AC = I, then B = C by applying the properties of matrix multiplication, associativity, and identity matrices. This result shows that a matrix A can have only one inverse.

Step-by-step solution

Multiply both equations by A on the left

First, we can simply multiply both equations by A on the left side to keep the equal relationship between them: \(\mathbf{A}(\mathbf{A B})=\mathbf{A}(\mathbf{I})\) and \(\mathbf{A}(\mathbf{A C})=\mathbf{A}(\mathbf{I})\)

Apply the associative property of matrix multiplication

The associative property of matrix multiplication states that \((\mathbf{A}\mathbf{B})\mathbf{C}=\mathbf{A}(\mathbf{B}\mathbf{C})\) for any matrices A, B, and C. Applying the associative property to the two equations above, we have: \((\mathbf{A A})\mathbf{B} = (\mathbf{A A})\mathbf{C}\)

Use the definition of the Inverse Matrix

According to the definition of the inverse matrix, \(\mathbf{A A}^{-1}=\mathbf{I}\). Since both \(\mathbf{A B}\) and \(\mathbf{A C}\) equal to identity matrix \(\mathbf{I}\), we can replace it: \(\mathbf{I}\mathbf{B} = \mathbf{I}\mathbf{C}\)

Use the Identity matrix property

The identity matrix I serves as the neutral element for matrix multiplication, meaning that for any matrix A, \(\mathbf{I A = A I = A}\). Applying this property to our equation, we have: \(\mathbf{B} = \mathbf{C}\)

Conclusion

We have shown that if there are two matrices B and C such that \(\mathbf{A B} = \mathbf{I}\) and \(\mathbf{A C} = \mathbf{I}\), then \(\mathbf{B} = \mathbf{C}\). This result establishes that a matrix A can have only one inverse, proving the statement in the exercise.

Key concepts

Associative Property of Matrix Multiplication

The associative property of matrix multiplication is a fundamental property that simplifies the process of working with matrices. It states that for any matrices

• A, B, and C,
• the equation is \[(AB)C = A(BC)\] holds.

This property assures us that even if the order of multiplication changes, the resultant matrix remains the same, as long as the parenthesis are used according to the equation above.
Understanding this property is very important when dealing with equations involving multiple matrices. It allows one to regroup matrices during multiplication, making calculations easier and potentially reducing computational errors.
In the exercise, this property was crucial in rearranging the matrix equation so that we could apply it correctly and move closer towards proving the identity: \( \mathbf{B} = \mathbf{C} \).

Identity Matrix

An identity matrix, often denoted as \( \mathbf{I} \), is a special type of matrix that acts as a multiplicative identity in matrix algebra, similar to the number one in basic arithmetic. For any matrix \( \mathbf{A} \), regardless of its size, multiplying it by the identity matrix yields:

• \( \mathbf{IA} = \mathbf{A} \)
• \( \mathbf{AI} = \mathbf{A} \)

The identity matrix is characterized by having 1’s along its main diagonal and 0’s everywhere else. This property is vital, especially when solving matrix equations, as seen in the exercise.
In our case, the identity matrix helped prove the statement \( \mathbf{A B} = \mathbf{I} \) and \( \mathbf{A C} = \mathbf{I} \) ultimately leading to \( \mathbf{B} = \mathbf{C} \). It is a powerful tool in matrix algebra, simplifying computations and allowing the derivation of important results like showing the uniqueness of an inverse matrix.

Inverse Matrix

An inverse matrix is the matrix which, when multiplied by the original matrix, yields the identity matrix. It's akin to finding a reciprocal in arithmetic, where multiplying a number by its reciprocal equates to one.

• For any square matrix \( \mathbf{A} \), if there exists a matrix \( \mathbf{B} \) such that \( \mathbf{AB} = \mathbf{BA} = \mathbf{I} \),

then \( \mathbf{B} \) is called the inverse of \( \mathbf{A} \). Not every matrix has an inverse, so those that do are referred to as invertible or non-singular.
In the exercise we explored, the inverse matrix concept was essential in proving that if \( \mathbf{A B} = \mathbf{I} \) and \( \mathbf{A C} = \mathbf{I} \), then \( \mathbf{B} = \mathbf{C} \). This proves the uniqueness of the inverse matrix, showing that there can be only one such matrix \( \mathbf{B} \) that operates as the inverse of \( \mathbf{A} \).
Understanding how to find and apply inverse matrices is key in solving many matrix-related problems, including systems of linear equations.

Matrix Multiplication Properties

Matrix multiplication is governed by a set of properties that dictate how matrices can be combined and manipulated to produce desired outcomes. These properties include:

• Associative Property: Allows regrouping of matrices without changing the product: \( (AB)C = A(BC) \).
• Non-Commutative: Unlike numbers, matrices cannot be freely reordered in multiplication: generally \( AB eq BA \).
• Distributive Property: Over addition, matrix multiplication distributes: \( A(B + C) = AB + AC \).

These properties underline all operations involving matrices. Understanding them is crucial to correctly executing and simplifying complex matrix equations.
In the context of our exercise, recognizing these properties helped confirm the dependency and uniqueness of the inverse. Pairing these principles with others like the identity matrix assisted in simplifying proof and ensuring logical progression when demonstrating mathematical results related to matrix inversion.