Problem 47 - Precalculus

Question

Determine whether \(B\) is the multiplicative inverse of \(A\) using \(A A^{-1}=I\). $$ A=\left[\begin{array}{rr} 3 & 1 \\ 1 & -2 \end{array}\right] \quad B=\left[\begin{array}{cc} \frac{2}{7} & \frac{1}{7} \\ \frac{1}{7} & -\frac{3}{7} \end{array}\right] $$

Step-by-Step Solution

Answer

Yes, matrix \(B\) is the multiplicative inverse of matrix \(A\).

Step 1: Understand the Problem

To determine if matrix \(B\) is the multiplicative inverse of matrix \(A\), we multiply the matrices \(A\) and \(B\) to see if the result is the identity matrix \(I\). The identity matrix for 2x2 matrices is \(I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).

Step 2: Set Up Matrix Multiplication

Multiply matrix \(A\) by matrix \(B\). We will calculate the product of two 2x2 matrices \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) and \(\begin{bmatrix} e & f \ g & h \end{bmatrix}\) using the formula: \(\begin{bmatrix} ae + bg & af + bh \ ce + dg & cf + dh \end{bmatrix}\).

Step 3: Perform the Matrix Multiplication

Multiply the corresponding entries:- First row, first column: \(3 \times \frac{2}{7} + 1 \times \frac{1}{7} = \frac{6}{7} + \frac{1}{7} = 1\).- First row, second column: \(3 \times \frac{1}{7} + 1 \times -\frac{3}{7} = \frac{3}{7} - \frac{3}{7} = 0\).- Second row, first column: \(1 \times \frac{2}{7} + (-2) \times \frac{1}{7} = \frac{2}{7} - \frac{2}{7} = 0\).- Second row, second column: \(1 \times \frac{1}{7} + (-2) \times -\frac{3}{7} = \frac{1}{7} + \frac{6}{7} = 1\).

Step 4: Check the Result

After multiplication, the result is \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\), which is the identity matrix \(I\). Therefore, \(B\) is indeed the multiplicative inverse of \(A\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a technique used to combine two matrices to produce a third matrix. In the context of determining matrix inverses, it is crucial because it helps verify if two matrices, when multiplied together, return the identity matrix. This relationship is foundational for understanding the mathematics of linear transformations.
For 2x2 matrices, matrix multiplication involves multiplying corresponding elements and summing up the results to fill each position of the resulting matrix. Given a matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) and a matrix \(B = \begin{bmatrix} e & f \ g & h \end{bmatrix}\), the resulting product \(AB\) can be calculated as follows:

• The element at the first row, first column is given by \(ae + bg\).
• The element at the first row, second column is given by \(af + bh\).
• The element at the second row, first column is given by \(ce + dg\).
• The element at the second row, second column is given by \(cf + dh\).

This operation requires each element of the rows of the first matrix to interact with each element of the columns of the second matrix, reflecting the sum of products for each entry. Understanding and mastering this pattern is essential for working with linear equations and transformations.

Identity Matrix

An identity matrix is a special kind of square matrix that plays a pivotal role in matrix operations. It is the "do nothing" component in matrix multiplication, akin to multiplying by one in regular arithmetic. The identity matrix is characterized by having 1's along its main diagonal (from top left to bottom right) and 0's elsewhere.
For a 2x2 identity matrix, the representation is:\[I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\] The identity matrix has a unique property in multiplication:

• When any matrix is multiplied by the identity matrix, the original matrix remains unchanged.
• Thus, if matrix \(A\) is multiplied by its inverse \(A^{-1}\) and the result is the identity matrix \(I\), \(B\) can be confirmed as the inverse of \(A\).

Finding the identity matrix through multiplication helps prove the correctness of an inverse, crucial for solving systems of equations and inverting transformations. The clear and predictable nature of the identity matrix makes it a fundamental tool in linear algebra.

2x2 Matrices

2x2 matrices are the simplest form of square matrices, consisting of two rows and two columns. Despite their simplicity, they hold great significance in linear algebra for modeling linear transformations such as scaling, rotating, and reflecting in two-dimensional space.
The general form of a 2x2 matrix is:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]Each entry of the matrix has its role and represents a component of the transformation being applied. These matrices are integral in computer graphics, physics, and engineering for simplifying and handling complex calculations in a digestible format.

• They support basic operations such as addition, subtraction, and importantly, multiplication.
• The concept of an inverse is particularly powerful, allowing one to "undo" a transformation.
• Whether you're multiplying matrices, finding determinants, or calculating inverses, 2x2 matrices serve as accessible models for understanding higher-dimensional linear algebra concepts.

By mastering 2x2 matrices, students build a necessary foundation for tackling larger and more complex systems in mathematics and applied sciences.