Question

Prove: If \(\mathbf{F}\) is invertible, then \(\mathbf{F}^{-1}\) is unique.

Short answer

Question: Prove that if a matrix 𝐅 is invertible, then its inverse, denoted by 𝐅^(-1), is unique. Answer: By proof by contradiction, we assume that there are two distinct inverses 𝐀 and 𝐁 for the matrix 𝐅. Using the identity matrix 𝐈, we multiply the first equation involving 𝐀 by 𝐁 and substitute the inverse property of 𝐁. We then simplify by removing the identity matrix, resulting in 𝐀 = 𝐁. This means that 𝐀 and 𝐁 must be the same matrix, and there cannot be two distinct inverses for the given matrix 𝐅. Therefore, the inverse of an invertible matrix is unique.

Step-by-step solution

Proof by contradiction

Let's suppose, for the sake of contradiction, that there are two distinct inverses of the matrix \(\mathbf{F}\), which we will call \(\mathbf{A}\) and \(\mathbf{B}\). In other words, both \(\mathbf{A}\) and \(\mathbf{B}\) satisfy the conditions for being the inverse of \(\mathbf{F}\): $$\mathbf{AF} = \mathbf{FA} = \mathbf{I}$$ and $$\mathbf{BF} = \mathbf{FB} = \mathbf{I}$$ where \(\mathbf{I}\) is the identity matrix.

Multiply both sides by \(\mathbf{B}\)

Let's take the first equation involving \(\mathbf{A}\) and multiply both sides on the right by \(\mathbf{B}\): $$\mathbf{AFB} = \mathbf{I}\mathbf{B}$$ Using the associativity property of matrix multiplication, we can rewrite the left side of the equation as: $$\mathbf{A(FB)} = \mathbf{IB}$$

Substitute the inverse property of \(\mathbf{B}\)

Now, we know from the second equation that \(\mathbf{FB} = \mathbf{I}\). So, we can substitute this into our current equation: $$\mathbf{A}\mathbf{I} = \mathbf{IB}$$

Simplify by removing the identity matrix

Since multiplying any matrix by the identity matrix results in the original matrix, we can now simplify the equation: $$\mathbf{A} = \mathbf{B}$$

Conclusion

Thus, if \(\mathbf{F}\) is invertible, then its inverse is unique. In this case, it means that \(\mathbf{A}\) and \(\mathbf{B}\) must be the same matrix, and there cannot be two distinct inverses for the given matrix \(\mathbf{F}\).

Key concepts

Invertible Matrix

An invertible matrix, also known as a non-singular matrix, is essentially a square matrix that possesses a unique inverse. The ability to have an inverse is a significant property, as it means that matrix equations involving the invertible matrix can be solved. For a matrix \( \mathbf{F} \) to be invertible, it must satisfy the condition that there exists another matrix \( \mathbf{F}^{-1} \) such that:

\[ \mathbf{FF^{-1}} = \mathbf{F^{-1}F} = \mathbf{I} \]
where \( \mathbf{I} \) is the identity matrix. It's the matrix equivalent of dividing numbers, where finding the multiplicative inverse allows for division. Just as a number multiplied by its reciprocal gives 1, an invertible matrix multiplied by its inverse gives the identity matrix.

An important aspect of the invertible matrix is the determinant. If the determinant of a square matrix is non-zero, the matrix is invertible. Otherwise, if the determinant is zero, it's a singular matrix and does not have an inverse, trapping equations in a limbo without unique solutions.

Identity Matrix

The identity matrix, denoted by \( \mathbf{I} \), is a special kind of square matrix with ones on the diagonal and zeros elsewhere. It's the equivalent of the number '1' in matrix operations. This means that when any matrix \( \mathbf{F} \) is multiplied by the identity matrix, \( \mathbf{F} \) remains unchanged, symbolically:

\[ \mathbf{FI} = \mathbf{IF} = \mathbf{F} \]
This property is crucial in proofs and computations involving matrices. In the context of invertible matrices, the identity matrix serves as the 'goal' to demonstrate that another matrix is indeed the inverse by showing it reproduces the identity matrix upon multiplication. Whenever students deal with matrix equations, it's essential to recognize the identity matrix as the benchmark for both starting and verifying the completion of matrix operations.

Proof by Contradiction

The method of proof by contradiction is a powerful logical process used to establish the truth of a statement. It starts by assuming the opposite of what you want to prove is true and then demonstrating that this assumption leads to a contradiction. The contradiction shows that the original assumption must be false, thus proving the initial statement was true. In our matrix problem, we apply this method by assuming there are two distinct inverses for the matrix \( \mathbf{F} \). For the matrices \( \mathbf{A} \) and \( \mathbf{B} \) to both be inverses of \( \mathbf{F} \) and yet be distinct from one another would violate the uniqueness that comes from the definition of an inverse. By showing that this assumption results in \( \mathbf{A} = \mathbf{B} \), we reach a contradiction because they were presumed distinct—effectively proving the inverse of a matrix is unique.

Matrix Multiplication

Understanding matrix multiplication is vital for working with matrices. Unlike multiplying numbers, matrix multiplication involves a set of rules for combining the rows and columns. For two matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix has dimensions that match the number of rows in the first matrix and the number of columns in the second matrix.

A key property of matrix multiplication is associativity, which was used in our example. Associativity means that the way matrices are grouped doesn't affect the product:

\[ (\mathbf{AB})\mathbf{C} = \mathbf{A}(\mathbf{BC}) \]
This can greatly simplify the task of proving certain matrix properties, such as the uniqueness of inverse, because you can 'move' the parentheses around without changing the product. Despite being associative, matrix multiplication is not commutative in general, meaning \( \mathbf{AB} \) does not necessarily equal \( \mathbf{BA} \). This lack of commutativity must be kept in mind when working with matrices.