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 \( I_3 \) is \( I_3 \) itself.

Step 1: Understand the Identity Matrix

The matrix given is an identity matrix, denoted by \( I_3 \), which is a 3x3 square matrix with ones on the diagonal and zeros elsewhere. The entries of \( I_3 \) are: \[ I_3 = \begin{bmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1\end{bmatrix} \]

Step 2: Recall the Property of Identity Matrices

An important property of identity matrices is that they are their own inverse. This means that multiplying an identity matrix with itself results in the same identity matrix: \( I_3 \cdot I_3 = I_3 \)

Step 3: Verify the Inverse Property

We know the inverse of a matrix \( A \) is \( A^{-1} \) such that the product \( AA^{-1} = I \). The identity matrix already satisfies this condition since: \[ I_3 \cdot I_3 = \begin{bmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1\end{bmatrix} \]This equals \( I_3 \), fulfilling the requirement for the inverse.

Key Concepts

Identity Matrix

An identity matrix is a special kind of matrix in linear algebra. It's denoted by the symbol \( I_n \), where \( n \) is the size of the square matrix. In the identity matrix, all the diagonal elements are ones, while all other elements are zeros. For example, a 3x3 identity matrix, \( I_3 \), simply looks like:

• Diagonal: All ones
• Non-diagonal: All zeros

This special arrangement means that the identity matrix behaves like the number 1 in regular multiplication, hence it is also sometimes referred to as the 'multiplicative identity' for matrices. When any matrix is multiplied by an identity matrix, the original matrix is obtained. This property is consistently true whether the identity matrix is on the left or right side of the multiplication. Understanding the characteristics of the identity matrix is fundamental as it lays the foundation for more complex topics like calculating the inverse of a matrix. Knowing that an identity matrix is its own inverse helps simplify many computations in linear algebra.

Matrix Properties

Matrix properties form the backbone of understanding matrix operations. One key property is the concept of an inverse. Not all matrices have inverses, but for those that do, this inverse \( A^{-1} \) satisfies the equation \( AA^{-1} = I \), where \( I \) is the identity matrix. This means multiplying a matrix by its inverse results in the identity matrix.

• Invertibility: For a matrix to have an inverse, it must be square (same number of rows and columns) and it must be non-singular, meaning its determinant is not zero.
• Unique Inverse: If a matrix has an inverse, it is unique. This means for any matrix \( A \), there exists only one inverse matrix \( A^{-1} \) that satisfies the condition of resulting in the identity matrix when multiplied by \( A \).

These properties are essential when solving systems of equations using matrices, as the inverse can be used to simplify complex matrix equations. Recognizing when these properties apply simplifies working with different kinds of matrices, especially when verifying if certain matrices will behave as expected in calculations.

Matrix Multiplication

Matrix multiplication is a core operation in linear algebra, and it differs substantially from regular numerical multiplication. Knowing how to multiply matrices is crucial for operations involving the identity matrix and finding inverses.

• Element-wise Calculation: To multiply two matrices \( A \) and \( B \), the number of columns in \( A \) must equal the number of rows in \( B \). This ensures that each element in the resulting matrix is calculated correctly.
• Associative Property: Matrix multiplication is associative, which means \((AB)C = A(BC)\). This property is very useful in long chains of matrix multiplications.
• Non-Commutative: Unlike regular numbers, matrices are generally not commutative under multiplication, meaning \( AB eq BA \). This property underscores the importance of the order in matrix operations.

When working with identity matrices, multiplying any matrix \( A \) by an identity matrix of appropriate size results in \( A \). This unique property is harnessed in various applications, including solving systems of linear equations. Understanding these multiplication rules is key to efficiently handling more advanced matrix computations in algebra.