Problem 50 - Algebra and Trigonometry

Question

Prove the following statement: If \(A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right]\) and \(a d-b c \neq 0\) then \(A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}{d} & {-b} \\ {-c} & {a}\end{array}\right]\) (Hint: Use the method of Example 2 on page 943 .)

Step-by-Step Solution

Answer

The given formula \( A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}{d} & {-b} \ {-c} & {a}\end{array}\right]\) was proven as the correct formula for the inverse of a 2x2 matrix \( A \). When Matrix \( A \) is multiplied by its supposed inverse, the resulting matrix is the identity matrix which is the definition of an inverse matrix.

Step 1: Understand the Definition of the Inverse of a Matrix

The inverse of a matrix \( A \) is another matrix, denoted as \( A^{-1} \), such that \( A \times A^{-1} = A^{-1} \times A = I \), where \( I \) is the identity matrix. In this case, the identity matrix is \( I = \left[\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right] \)

Step 2: Multiply Matrix A by its Inverse

Multiply matrix \( A \) by its supposed inverse. \( A \times A^{-1} =\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \times \frac{1}{{a d-b c}}\left[\begin{array}{rr} d & -b \\ -c & a \end{array}\right] \). This will result in a new 2x2 matrix.

Step 3: Perform the Matrix Multiplication

The result of the multiplication is \(\frac{1}{{a d-b c}}\left[\begin{array}{ll} (a \cdot d - b \cdot (-c)) & (a \cdot (-b) - b \cdot a) \\ (c \cdot d - d \cdot (-c)) & ((-c) \cdot b + d \cdot a) \end{array}\right] = \frac{1}{{a d-b c}}\left[\begin{array}{rr} (a \cdot d + b \cdot c) & 0 \\ 0 & (a \cdot d + b \cdot c) \end{array}\right] = \left[\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right] = I \). Note that in order for this to be possible, \( a d - b c \neq 0 \), which is given in the problem.

Step 4: Confirm the Resulting Matrix is the Identity Matrix

The result of the matrix multiplication is the identity matrix. This confirms that the given formula for the inverse of matrix \( A \) is indeed correct, as the definition of the inverse of matrix \( A \) is satisfied.

Key Concepts

Determinant

To understand the inverse of a matrix, we first introduce the concept of a determinant. The determinant is a scalar value that is a function of a square matrix. It provides crucial insights into the properties of the matrix, such as whether the matrix is invertible. For a 2x2 matrix, the determinant can be calculated using the formula:

• Given a matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), its determinant, denoted as \( \text{det}(A) \), is \( ad - bc \).

In the context of the exercise, the condition \( ad - bc eq 0 \) ensures that matrix \( A \) is invertible. If the determinant were zero, the matrix would not have an inverse and could not be used in solving linear equations. Understanding how to calculate and interpret determinants is foundational when working with matrices.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix. It is a square array consisting of two rows and two columns. Structurally, it looks like this:

• \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \)

Each element in the matrix has a specific position. For example, "\( a \)" is at the top-left, "\( b \)" is at the top-right, "\( c \)" is at the bottom-left, and "\( d \)" is at the bottom-right. In matrix operations like addition, multiplication, and finding inverses, the arrangement and values in these positions greatly affect the outcome. When studying simple matrix operations, beginning with a 2x2 matrix also serves as an excellent introduction to understanding larger matrices.

Identity Matrix

The identity matrix plays a vital role in matrix operations, much like the number 1 in multiplication for real numbers. For a 2x2 matrix, the identity matrix is:

• \( I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \)

This matrix possesses a unique property where any matrix multiplied by it results in the original matrix. Mathematically, \( A \times I = A \) and \( I \times A = A \). In the process of finding the matrix inverse, confirming that multiplying a matrix and its inverse results in the identity matrix \( I \) is necessary to ensure the inverse has been correctly calculated.

Matrix Multiplication

Matrix multiplication is fundamental when solving systems of equations, particularly when dealing with matrices like inverses. Multiplying matrices requires careful alignment of rows and columns. Here's how to perform matrix multiplication with a 2x2 matrix:

• Multiplying \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) by \( B = \begin{pmatrix} e & f \ g & h \end{pmatrix} \)
• The resultant matrix \( C = A \times B \) will be another 2x2 matrix:

\[ C = \begin{pmatrix} ae + bg & af + bh \ ce + dg & cf + dh \end{pmatrix} \]Each element in the resulting matrix is determined by summing the products of corresponding elements in the row of the first matrix and the column of the second matrix. Practicing matrix multiplication with smaller matrices like 2x2s helps build intuition for more complex matrix operations.