Problem 58 - Discrete Mathematics with Applications

Question

Prove each. 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 can denote an unknown matrix as \(B = \left(A^{-1}\right)^{-1}\). By multiplying both sides by \(A^{-1}\) on the right and using the definition of an inverse matrix and the matrix multiplication property, we can show that \(BA^{-1} = A^{-1}\left(A^{-1}\right)^{-1}\). Upon simplification, we find that \(B=I\), indicating that \(\left(A^{-1}\right)^{-1}=A\).

Step 1: Introduce the unknown matrix

Let us denote the unknown matrix as \(B\), such that \(B = \left(A^{-1}\right)^{-1}\). We need to prove that \(B = A\).

Step 2: Multiply both sides by \(A^{-1}\)

Multiply both sides of the equation \(B = \left(A^{-1}\right)^{-1}\) by \(A^{-1}\) on the right: \(BA^{-1} = \left(A^{-1}\right)^{-1}A^{-1}\)

Step 3: Use the definition of an inverse matrix

By the definition of an inverse matrix, since \(A\) is invertible, we have that \(AA^{-1}=I\). So, we can rewrite the product \(\left(A^{-1}\right)^{-1}A^{-1}\): \(BA^{-1} = A^{-1}\left(A^{-1}\right)^{-1}\)

Step 4: Invoke the matrix multiplication property

Since matrix multiplication is associative, we can write: \(B(AA^{-1}) = (BA)A^{-1}\) Now we can use the definition of the inverse matrix again: \(BI = BA^{-1}\)

Step 5: Simplify

Since \(BI = B\), we have: \(B = BA^{-1} = A^{-1}\left(A^{-1}\right)^{-1}\) Recall that our goal was to show that \(B=A\). From step 4, we showed that \(B = A^{-1}\left(A^{-1}\right)^{-1}\). By the definition of the inverse matrix, we can substitute \(A\): \(B=AA^{-1}\) And since \(AA^{-1}=I\), we have: \(B=I\) Thus, we have shown that if \(A\) is an invertible matrix, then \(B = \left(A^{-1}\right)^{-1}=A\).

Key Concepts

Invertible Matrix

An invertible matrix, often called a non-singular or non-degenerate matrix, is one that possesses an inverse. In simple terms, a matrix is invertible if there exists another matrix that, when multiplied with it, results in the identity matrix. The identity matrix is essentially the matrix world's version of the number 1 in arithmetic.
Essentially, for a matrix \(A\), if there exists a matrix \(B\) such that \(AB = BA = I\), where \(I\) is the identity matrix, then \(A\) is considered invertible and \(B\) is its inverse, denoted as \(A^{-1}\). Not all matrices are invertible; a matrix must be square (having the same number of rows and columns), and its determinant must not be zero to be invertible.
Checking whether a matrix is invertible is crucial in many mathematical computations and applications, including solving systems of linear equations and transforming geometrical objects.

Inverse Matrix Properties

Inverse matrices have interesting properties that are essential when working with them. These properties simplify complex matrix calculations once you grasp them. One fundamental property is that the inverse of the inverse matrix is the original matrix itself, symbolized as \((A^{-1})^{-1} = A\). This property is what the exercise is about proving.
Additionally, inverse matrices follow these rules:

• \((AB)^{-1} = B^{-1}A^{-1}\), implying the inverse of a product is the reverse product of the inverses.
• If \(A\) and \(B\) are invertible matrices of the same size, then their product \(AB\) is also invertible.
• If \(A\) is invertible, then \(A^{-1}\) is also invertible, and \(A\) becomes its inverse.

Understanding these properties is crucial for manipulating matrices and comprehending how inverse matrices function within larger mathematical systems.

Matrix Multiplication

Matrix multiplication is an operation that takes two matrices and produces another matrix. The rule for multiplying matrices is that the number of columns in the first matrix must equal the number of rows in the second matrix. The resulting matrix has dimensions based on the rows of the first and columns of the second.
Matrix multiplication is not commutative, which means for \(A\) and \(B\), generally \(AB eq BA\). However, it is associative, meaning \((AB)C = A(BC)\). It's also distributive over addition, so you can write \(A(B + C) = AB + AC\). Being able to multiply matrices correctly is crucial for operations such as transforming coordinates or solving linear equations using matrix forms.

• To multiply two matrices, match the corresponding elements and sum up the products.
• Ensure the order of multiplication is correct, as switching the order usually changes the product.

This operation, while seemingly complex at first, becomes more manageable and intuitive with practice.

Associative Property of Matrix Multiplication

The associative property is a key characteristic of matrix multiplication, ensuring that when you are multiplying several matrices together, the order in which you perform the multiplications doesn't affect the final product.
Simply put, if you have three matrices \(A\), \(B\), and \(C\), it doesn't matter whether you first multiply \(A\) and \(B\) and then multiply the result by \(C\), or if you multiply \(B\) and \(C\) first and then \(A\), because \((AB)C = A(BC)\).
This property is especially useful when working with larger matrices or in more complex algebraic manipulations because it allows flexibility in grouping numbers or rearranging computations. It emphasizes focusing on other properties and rules such as invertibility and order of multiplication.
Understanding the associative property helps in efficiently simplifying and solving matrix equations and computations.