Problem 58 - Discrete Mathematics with Applications

Question

If \(A\) is an invertible matrix, then \(\left(A^{-1}\right)^{-1}=A\).

Step-by-Step Solution

Answer

To prove that if A is an invertible matrix, then \(\left(A^{-1}\right)^{-1}=A\), we first recall that an invertible matrix A has an inverse \(A^{-1}\) such that \(AA^{-1}=A^{-1}A=I\), where I is the identity matrix. We then multiply both sides of \((A^{-1})^{-1} = A\) by \(A^{-1}\), which results in the equation \(A^{-1}(A^{-1})^{-1} = I\). This equation shows that there exists a matrix \((A^{-1})^{-1}\) that is the inverse of \(A^{-1}\). Therefore, we have proved that \(\left(A^{-1}\right)^{-1}=A\).

Step 1: Definition of an Invertible Matrix and its Inverse

An invertible matrix A is a matrix that has an inverse matrix, A^-1, such that their product, in any order, will result in the identity matrix, I: \[AA^{-1}=A^{-1}A=I\]

Step 2: Multiply A^-1 by the Matrix A

We want to prove that \((A^{-1})^{-1} = A\). To do this, we will multiply both sides of the equation by A^-1: \[A^{-1}(A^{-1})^{-1} = AA^{-1}\]

Step 3: Observe the Identity Matrix

Recall that the product of a matrix and its inverse is an identity matrix (I). Since \(AA^{-1} = I\), the equation becomes: \[A^{-1}(A^{-1})^{-1} = I\]

Step 4: Definition of Inverse of A^{-1}

Now, let's understand what \((A^{-1})^{-1}\) signifies. It means that there exists a matrix such that the product of itself and A^-1 results in the identity matrix. In other words, the equation from step 3 fulfills this condition. Since \(A^{-1}(A^{-1})^{-1} = I\), the matrix \((A^{-1})^{-1}\) is indeed the inverse of A^-1.

Step 5: Arriving at the Proof

Since we've shown that the matrix \((A^{-1})^{-1}\) is the inverse of A^-1, and by step 3, we've established that the product of A and A^-1 gives us the identity matrix, we can now conclude that: \[\left(A^{-1}\right)^{-1}=A\] This proof shows that if A is an invertible matrix, then the inverse of its inverse, A^-1, equals the original matrix, A.

Key Concepts

Invertible Matrix

An invertible matrix, sometimes called a non-singular or non-degenerate matrix, is a square matrix that has an inverse. For a matrix to be invertible, there must exist another matrix, called the inverse matrix, such that their product is the identity matrix.
This concept is crucial in linear algebra because it allows us to "undo" the effects of a matrix using its inverse. The mathematical relationship that defines an invertible matrix is:

• For a matrix \(A\) to be invertible, there must exist a matrix \(A^{-1}\) such that \(AA^{-1} = A^{-1}A = I\).
• The identity matrix \(I\) is the matrix representation where all the diagonal elements are 1, and all off-diagonal elements are 0.

A matrix that does not meet this condition is considered singular and does not have an inverse.

Inverse Matrix

The inverse of a matrix \(A\) is denoted by \(A^{-1}\) and is the matrix that, when multiplied by \(A\), yields the identity matrix. Finding an inverse is a process applicable only to square matrices, and not all matrices have inverses.
Here's a brief summary of the key attributes of inverse matrices:

• Only square matrices (matrices with the same number of rows and columns) can have inverses.
• The product of a matrix and its inverse, in any order, results in the identity matrix: \(AA^{-1} = A^{-1}A = I\).
• If \(A\) is invertible, formulas or methods like Gaussian elimination, the adjugate method, or using row reduction can be used to find \(A^{-1}\).

Understanding the inverse is foundational for solving systems of equations, where the inverse provides a method to solve \(AX = B\) by computing \(X = A^{-1}B\). But remember, if \(A^{-1}\) exists, then \(AX = 0\) implies \(X = 0\).

Identity Matrix

An identity matrix is central to the concept of invertible matrices. It is a special type of square matrix where all elements on the principal diagonal are 1, and all other elements are 0, making it a matrix analog of the number 1 in arithmetic.
The key properties of the identity matrix are:

• When any matrix is multiplied by the identity matrix, it remains unchanged: \(AI = IA = A\).
• The identity matrix serves as the "neutral" element in matrix multiplication.
• For a \(3 \times 3\) matrix, the identity matrix would look like this: \[I = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix}\]

In practical applications, verifying that a particular matrix is indeed an identity matrix can confirm that two matrices are inverses of each other.

Proof Techniques

Proof techniques in mathematics are crucial for verifying concepts like matrix inversion. Understanding how to construct a proof verifies the truths of algebraic statements. One common technique used is direct proof, particularly beneficial in the context of invertible matrices.
For matrix inversion, the step-by-step approach generally includes:

• Starting with known theorems or established mathematics principles—such as the fact that multiplying a matrix by its inverse yields the identity matrix.
• Employing algebraic manipulations, like moving terms and multiplying both sides of an equation by a matrix or its inverse.
• Reaching a conclusion that directly follows logically from the initial assumptions and calculations.

When showing that \((A^{-1})^{-1} = A\), the proof connects through definitions and properties that hinge on knowing the behavior of inverses and identity matrices. Through these steps, one ensures solutions are not just observed but understood for deeper math comprehension.