Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-2: Show that \(B\) is the inverse of \(A\). $$ A=\left[\begin{array}{ll} 5 & 7 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 3 & -7 \\ -2 & 5 \end{array}\right] $$

Step-by-Step Solution

Answer

Matrix B is the inverse of Matrix A.

Step 1: Understand the Concept of Inverse

The inverse of a matrix \(A\), denoted as \(A^{-1}\), is a matrix that, when multiplied by \(A\), results in the identity matrix. The identity matrix \(I\) for a 2x2 matrix is \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).

Step 2: Multiply Matrix A with Matrix B

To check if \(B\) is the inverse of \(A\), compute the product of \(A\) and \(B\). First, multiply the first row of \(A\) by the first column of \(B\) to get the first element of the result matrix: \(5 \times 3 + 7 \times (-2) = 15 - 14 = 1\).

Step 3: Continue Matrix Multiplication for Remaining Elements

Continue multiplying as described:- First row of \(A\) with the second column of \(B\): \(5 \times (-7) + 7 \times 5 = -35 + 35 = 0\).- Second row of \(A\) with the first column of \(B\): \(2 \times 3 + 3 \times (-2) = 6 - 6 = 0\).- Second row of \(A\) with the second column of \(B\): \(2 \times (-7) + 3 \times 5 = -14 + 15 = 1\).

Step 4: Construct the Product Matrix

From the computations:- First row: \([1, 0]\)- Second row: \([0, 1]\)Combine these into the identity matrix: \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).

Step 5: Verify if the Product is the Identity Matrix

Since the product of \(A\) and \(B\) is \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\), this confirms that \(B\) is indeed the inverse of \(A\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra where you multiply two matrices to produce a new matrix. This process involves a systematic approach:

• Take each row of the first matrix.
• Multiply it with each column of the second matrix.
• Sum the products to get the entries of the resulting matrix.

For example, when multiplying two 2x2 matrices, you'll perform operations on each of the four possible (row, column) pairs. Each entry in the resulting matrix is calculated by multiplying elements from the corresponding row of the first matrix and column of the second matrix, then adding them up. Getting the hang of matrix multiplication is crucial because it is used to determine many important matrix-related results, such as checking if one matrix is the inverse of another.

Identity Matrix

The identity matrix is a special type of square matrix that acts as a multiplicative identity in matrix operations, much like the number 1 in regular arithmetic.

• For any matrix size, the identity matrix consists of 1's on the diagonal, from the top left corner to the bottom right.
• All other elements are 0.
• For a 2x2 matrix, it looks like this: \[\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]

When a matrix is multiplied by its identity matrix, the original matrix remains unchanged. In terms of inverses, if you find a matrix product that equals the identity matrix, it indicates that one of those matrices is the inverse of the other.

2x2 Matrix

A 2x2 matrix is a type of matrix with two rows and two columns. These matrices are among the simplest and are often used in introductory linear algebra to illustrate core concepts.Understanding a 2x2 matrix involves knowing:

• The general layout: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]
• How to perform operations like addition, subtraction, and multiplication with other matrices of the same size.
• The conditions under which a 2x2 matrix has an inverse. Specifically, a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix} \) has an inverse if the determinant \((ad - bc) eq 0\).
• The computation of inverses, where a matrix can be inverted by a specific formula once the determinant is established as non-zero.

Matrix Theory

Matrix theory is a branch of mathematics focused on the study of matrices. Matrices are versatile tools used to solve various mathematical problems, particularly in systems of linear equations.

• Core ideas of matrix theory include understanding how matrices can model linear transformations and systems.
• The theory explores various operations such as multiplication, transposition, and inversion.
• It also studies matrix properties like determinants, rank, eigenvalues, and eigenvectors, which have applications in fields like physics, computer science, and statistics.

Matrix theory emphasizes the importance of finding matrix inverses. An invertible (or non-singular) matrix is one that possesses an inverse, which can simplify solving linear equations by allowing matrix equations to be directly solved through methods like matrix division. Understanding these concepts deepens comprehension of complex systems and transformations, providing powerful tools for both theoretical and applied mathematics.