Problem 50 - Calculus for Biology and Medicine

Question

Find the inverse (if it exists) of $$ I_{3}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the identity matrix \(I_3\) is itself, \(I_3\).

Step 1: Understanding the Identity Matrix

To find the inverse of a matrix, we need to understand what we are working with. The given matrix is the identity matrix, denoted as \(I_3\). It is a 3x3 matrix with 1s on the diagonal and 0s elsewhere.

Step 2: Properties of the Identity Matrix

The identity matrix is unique because multiplying it with any square matrix of the same size results in the original matrix. It serves as a neutral element in matrix multiplication.

Step 3: Inverse of the Identity Matrix

The inverse of a matrix \(A\) is another matrix \(A^{-1}\) such that \(AA^{-1} = I\) and \(A^{-1}A = I\). Since the given matrix is an identity matrix, it is important to note that the identity matrix is its own inverse.

Step 4: Verification of Inverse Property

To verify, we can check if \(I_3 \cdot I_3 = I_3\). Performing the multiplication of \(I_3\) with itself results in:\[\begin{bmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{bmatrix}\cdot \begin{bmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{bmatrix}\]This shows \(I_3\) is indeed its own inverse.

Key Concepts

Identity Matrix

An identity matrix is a special type of square matrix in which all the elements on the diagonal from the top left to the bottom right are ones, and all other elements are zeros. For example, the 3x3 identity matrix is denoted as \( I_3 \) and looks like this:

• Diagonal elements: 1
• Non-diagonal elements: 0

This structure allows the identity matrix to act as a neutral element in matrix operations, much like the number one is for multiplication among real numbers. When any square matrix is multiplied by an identity matrix of the same order, the original matrix remains unchanged. This property is crucial in many areas of linear algebra, including inverting matrices. The identity matrix is its own inverse, meaning that multiplying it by itself results in itself, demonstrating its unique properties.

Matrix Multiplication

Matrix multiplication involves multiplying corresponding elements and summing them to produce elements in a new matrix. For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. The steps for multiplying two matrices are as follows:

• Find the dot product of rows from the first matrix and columns from the second matrix.
• Add up the products to form the elements of the result matrix.

Consider multiplying a 3x3 matrix by \( I_3 \):

• The original matrix is unaffected, proving that the identity matrix acts as a multiplicative identity.
• For instance, \( I_3 \times I_3 = I_3 \).

The identity matrix demonstrates how matrix multiplication respects what's called a neutral element, a crucial part of understanding matrix algebra.

Properties of Matrices

Matrices have several properties that dictate how they interact with one another. Understanding these properties helps when dealing with matrix operations, such as finding the inverse or performing matrix multiplication. Here are some essential properties:

• Associative property: \((AB)C = A(BC)\)
• Commutative property: This generally does not hold for matrix multiplication, meaning \( AB eq BA \) in most cases.
• Distributive property: \(A(B+C) = AB + AC\) and \((A+B)C = AC + BC\)

These properties reveal why certain results happen when matrices are multiplied or handled in algebraic expressions. For example, knowing that the identity matrix is its own inverse relies on recognizing these fundamental properties and applying them to test and verify matrix algebra outcomes.