Problem 47 - Calculus for Biology and Medicine

Question

$$ A=\left[\begin{array}{rr} -1 & 1 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -2 \\ 3 & 2 \end{array}\right] $$ Show that \(\left(A^{-1}\right)^{-1}=A\)

Step-by-Step Solution

Answer

The inverse of the inverse of matrix \( A \) is the matrix itself: \((A^{-1})^{-1} = A\).

Step 1: Understand the Problem

We are given two matrices \( A \) and \( B \), but we need to focus on \( A \). The task is to show that the inverse of the inverse of matrix \( A \) returns the original matrix \( A \). This means we need to prove that \((A^{-1})^{-1} = A\).

Step 2: Recall Matrix Inverse Definitions

By definition, for any invertible matrix \( A \), its inverse \( A^{-1} \) is the matrix such that \( A \times A^{-1} = I \), where \( I \) is the identity matrix. Also, there's a property stating that the inverse of an inverse matrix \((A^{-1})^{-1}\) is equal to the original matrix \( A \).

Step 3: Compute \( A^{-1} \)

The inverse of a 2x2 matrix \( A = \left[\begin{array}{rr}-1 & 1 \ 2 & 3\end{array}\right] \) is given by \( A^{-1} = \frac{1}{ad - bc} \left[\begin{array}{rr}d & -b \ -c & a\end{array}\right] \). Here, \( a = -1 \), \( b = 1 \), \( c = 2 \), \( d = 3 \). So, \( ad - bc = (-1)(3) - (1)(2) = -3 - 2 = -5 \). This leads to: \[ A^{-1} = \frac{1}{-5} \left[\begin{array}{rr}3 & -1 \ -2 & -1\end{array}\right] = \left[\begin{array}{rr}-\frac{3}{5} & \frac{1}{5} \ \frac{2}{5} & \frac{1}{5}\end{array}\right] \]

Step 4: Confirm the Identity \((A^{-1})^{-1} = A\)

We just need to apply the definition: the inverse of the inverse of a matrix \( A \) is the matrix itself. Therefore, without even calculation, by the properties mentioned in Step 2, \((A^{-1})^{-1} = A\).

Step 5: Verify by Matrix Multiplication

To satisfy our understanding, let's perform the multiplication of \( A \) and \( A^{-1} \): \[ A \times A^{-1} = \left[\begin{array}{rr}-1 & 1 \ 2 & 3\end{array}\right] \times \left[\begin{array}{rr}-\frac{3}{5} & \frac{1}{5} \ \frac{2}{5} & \frac{1}{5}\end{array}\right] = \left[\begin{array}{rr}\frac{3}{5}+\frac{2}{5} & -\frac{1}{5}+\frac{1}{5} \ -\frac{6}{5}+\frac{6}{5} & \frac{4}{5}+\frac{3}{5}\end{array}\right] = \left[\begin{array}{rr}1 & 0 \ 0 & 1\end{array}\right] \] This confirms that multiplying \( A \) by \( A^{-1} \) gives the identity matrix, hence, \((A^{-1})^{-1} = A\).

Key Concepts

2x2 Matrix

In mathematics, especially in linear algebra, a matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. A 2x2 matrix is a simple, yet fundamental type of matrix, as it consists of two rows and two columns, making it a total of four elements. Understanding a 2x2 matrix is crucial because it serves as the cornerstone for more complex matrix operations in higher dimensions. To illustrate, a general 2x2 matrix can be written as:

• \( A = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \)

Here, "a", "b", "c", and "d" are elements of the matrix.
A remarkable feature of a 2x2 matrix is that its determinant, crucial for calculating the inverse, can be easily computed. For the matrix \( A \), the determinant is given by \( ad - bc \). This determinant tells us if the matrix is invertible: the matrix has an inverse only if the determinant is not zero. This property is essential for solving various problems in linear algebra.

Linear Algebra

Linear algebra is a vital area of mathematics dealing with vectors, matrices, and linear transformations. It has applications in various fields such as engineering, physics, computer science, and economics. At its core, linear algebra provides the tools needed to solve systems of linear equations, perform transformations, and work with vector spaces.One of the key concepts in linear algebra is an

• Inverse of a Matrix: The inverse of a matrix \( A \) is denoted as \( A^{-1} \) and exists only if \( A \) is a square matrix with a non-zero determinant. The product of a matrix and its inverse results in the identity matrix, \( I \), which acts as the multiplicative identity in matrix operations.

Understanding these operations is essential for solving real-world problems, such as network analysis, image processing, and optimization tasks.
By mastering the basics of linear algebra, including properties of matrices and their inverses, students can tackle complex mathematical challenges with confidence.

Inverse Properties

Inverse properties in matrices are fundamental concepts that allow us to understand how matrices behave under multiplication. These properties not only help in solving equations but also foster a deeper comprehension of matrix operations.Some important inverse properties include:

• Inverse of the Inverse: For any invertible matrix \( A \), the inverse of its inverse is the matrix itself, expressed as \( (A^{-1})^{-1} = A \). This property demonstrates the symmetry and consistency within matrix operations.
• Identity Matrix: The matrix \( A \) multiplied by its inverse \( A^{-1} \) results in the identity matrix \( I \), meaning \( A \times A^{-1} = I \). The identity matrix is akin to the number 1 in regular multiplication, leaving the original matrix unchanged when multiplied.

Understanding these inverse properties is crucial for students exploring linear algebra, as they form the backbone of many theoretical and practical applications, from calculating unknown variables in equations to computer graphics rendering.
Thus, recognizing and applying inverse properties is indispensable for anyone delving into the world of linear matrix equations.