Problem 6 - Precalculus with Limits

Question

In Exercises 5-12, show that \(B\) is the inverse of \(A\). \(A = \left[ \begin{array}{r} 1 & -1 \\ -1 & 2 \end{array} \right]\), \(B = \left[ \begin{array}{r} 2 && 1 \\ 1 && 1 \end{array} \right]\).

Step-by-Step Solution

Answer

Matrix \(B\) is indeed the inverse of matrix \(A\) as multiplication in both orders produced the identity matrix.

Step 1: Multiplying A and B

Begin by performing the operation AB. This means each element of the resulting matrix is the sum of the products of corresponding elements from the rows of \(A\) and columns of \(B\). Calculate: \( \left[ \begin{array}{c} (1*2) + (-1*1) \ (-1*2) + (2*1) \ (1*1) + (-1*1) \ (-1*1) + (2*1) \end{array} \right] = \left[ \begin{array}{c} 1 \ 0 \ 0 \ 1 \end{array} \right] \). Hence AB = \(I\), where \(I\) is the identity matrix.

Step 2: Multiplying B and A

Next perform the operation BA. Compute: \( \left[ \begin{array}{c} (2*1) + (1*-1) \ (1*1) + (1*2) \ (2*-1) + (1*1) \ (1*-1) + (1*1) \end{array} \right] = \left[ \begin{array}{c} 1 \ 0 \ 0 \ 1 \end{array} \right] \). Therefore BA also equals \(I\).

Step 3: Confirming Inverse

Since AB = BA = \(I\), according to the definition of an inverse matrix, \(B\) is indeed the inverse of \(A\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in matrix algebra. When we multiply two matrices, we are essentially combining information from two different systems into one cohesive structure. Here’s how it works for matrices:

• Order matters: Matrix multiplication is not commutative, which means that \(A \times B eq B \times A\) in general.
• Dimensionality: When multiplying matrices, ensure the number of columns in the first matrix equals the number of rows in the second. If \(A\) is an \(m \times n\) matrix, and \(B\) is an \(n \times p\) matrix, the result will be an \(m \times p\) matrix.
• Element computation: To find each element of the resulting matrix, take the dot product of the corresponding row of the first matrix and the column of the second matrix.

In the exercise, multiplying \(A\) and \(B\) as well as \(B\) and \(A\) leads to the identity matrix, which is crucial for showing that these two matrices are inverses of each other.

Identity Matrix

The identity matrix is a special matrix in linear algebra, akin to the number 1 in multiplication. It plays a critical role as it maintains the properties of elements in matrix multiplication.

• Definition: An identity matrix is a square matrix with ones on the main diagonal and zeroes elsewhere. For a 2x2 matrix, it looks like this: \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).
• Multiplicative property: When you multiply any matrix by an identity matrix of compatible dimensions, the original matrix remains unchanged. Hence, \( A \times I = A \) and \( I \times A = A \).
• Determining inverses: If a matrix \(B\) is the inverse of matrix \(A\), then their product in either order results in the identity matrix, i.e., \( A \times B = I \) and \( B \times A = I \).

In the problem given, both \(A \times B\) and \(B \times A\) yield the identity matrix, confirming that \(B\) is indeed the inverse of \(A\).

Matrix Algebra

Matrix algebra encompasses various operations involving matrices, boosting our computational efficiency. This branch of mathematics includes operations like addition, subtraction, multiplication, and finding inverses.

• Inverse matrices: The inverse of a matrix \(A\) is another matrix \(B\) such that \(A \times B = I\) and \(B \times A = I\), where \(I\) is the identity matrix. Not all matrices have inverses; a matrix must be square and have a non-zero determinant to have one.
• Matrix properties: Working with matrices helps solve systems of equations, transform spaces, and analyze complex structures. Matrices can simplify operations that otherwise could be calculation-intensive.
• Importance in solving equations: Inverse matrices are used to solve linear equations of the form \(AX = B\). By multiplying both sides of the equation by \(A^{-1}\), the inverse of \(A\), we obtain \(X = A^{-1}B\).

The use of matrix algebra in this exercise is to identify \(B\) as the inverse of \(A\) and establish the relationship through matrix multiplication until the identity matrix is achieved.