Problem 4 - Precalculus: Mathematics for Calculus

Question

Verifying the Inverse of a Matrix Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A\) \(A=\left[\begin{array}{ll}2 & -3 \\ 4 & -7\end{array}\right] \quad B=\left[\begin{array}{ll}\frac{7}{2} & -\frac{3}{2} \\ 2 & -1\end{array}\right]\)

Step-by-Step Solution

Answer

Both AB and BA equal the identity matrix, confirming B is the inverse of A.

Step 1: Calculate the Product AB

To verify if matrix \( B \) is the inverse of matrix \( A \), we first calculate the product \( AB \), where: \[ A = \begin{bmatrix} 2 & -3 \ 4 & -7 \end{bmatrix}, B = \begin{bmatrix} \frac{7}{2} & -\frac{3}{2} \ 2 & -1 \end{bmatrix} \]We find the product using the following formula for matrix multiplication: \( AB = \begin{bmatrix} (2 \times \frac{7}{2} + (-3) \times 2) & (2 \times -\frac{3}{2} + (-3) \times -1) \ (4 \times \frac{7}{2} + (-7) \times 2) & (4 \times -\frac{3}{2} + (-7) \times -1) \end{bmatrix} \)Calculating each term, we have:\( \begin{align*}2 \times \frac{7}{2} + (-3) \times 2 & = 7 - 6 = 1, \2 \times -\frac{3}{2} + (-3) \times -1 & = -3 + 3 = 0, \4 \times \frac{7}{2} + (-7) \times 2 & = 14 - 14 = 0, \4 \times -\frac{3}{2} + (-7) \times -1 & = -6 + 7 = 1\end{align*} \)Thus, \( AB = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), which is the identity matrix.

Step 2: Calculate the Product BA

Now calculate the product \( BA \) where:\[ A = \begin{bmatrix} 2 & -3 \ 4 & -7 \end{bmatrix},B = \begin{bmatrix} \frac{7}{2} & -\frac{3}{2} \ 2 & -1 \end{bmatrix} \]Using the formula:\( BA = \begin{bmatrix} (\frac{7}{2} \times 2 + (-\frac{3}{2}) \times 4) & (\frac{7}{2} \times -3 + (-\frac{3}{2}) \times -7) \ (2 \times 2 + (-1) \times 4) & (2 \times -3 + (-1) \times -7) \end{bmatrix} \)Calculating each term, we find:\( \begin{align*}\frac{7}{2} \times 2 + (-\frac{3}{2}) \times 4 & = 7 - 6 = 1, \\frac{7}{2} \times -3 + (-\frac{3}{2}) \times -7 & = -\frac{21}{2} + \frac{21}{2} = 0, \2 \times 2 + (-1) \times 4 & = 4 - 4 = 0, \2 \times -3 + (-1) \times -7 & = -6 + 7 = 1\end{align*} \)Thus, \( BA = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), which is the identity matrix.

Step 3: Conclusion on Inverse Verification

Since both \( AB \) and \( BA \) resulted in the identity matrix \( I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), matrix \( B \) is indeed the inverse of matrix \( A \). Therefore, we have verified that \( B = A^{-1} \).

Key Concepts

Matrix Multiplication

Matrix multiplication is the process of multiplying two matrices together to form a new matrix. This operation is essential in many fields of mathematics and applications, especially in verifying inverse matrices. The method involves a systematic process where rows of the first matrix are multiplied by the columns of the second matrix. If you have matrices \( A \) and \( B \) with dimensions defined such that multiplication is possible, the resulting matrix \( AB \) will have dimensions that reflect the rows of the first matrix and columns of the second.

• Each element of the resulting matrix is obtained by multiplying the corresponding elements of the row and column that intersect at a specific position.
• Matrix multiplication is not commutative, meaning \( AB \) does not necessarily equal \( BA \).

For our exercise, matrix multiplication allows us to check if multiplying matrix \( A \) and matrix \( B \) gives us the identity matrix, thus indicating that \( B \) is indeed the inverse of \( A \). Understanding this operation is crucial as it builds the foundation for working with more complex equations in matrix algebra.

Identity Matrix

The identity matrix is a special kind of matrix that serves as the multiplicative identity in matrix algebra, much like the number 1 in arithmetic. It has a very simple structure: all elements along the diagonal are 1, and all other elements are 0.

• For a 2x2 identity matrix, it looks like \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).
• When any matrix is multiplied by the identity matrix of appropriate size, the original matrix remains unchanged.

This matrix is crucial when verifying inverses, because if the product of two matrices results in the identity matrix, then they are inverses of each other. In our given problem, both \( AB \) and \( BA \) equals the identity matrix, verifying that \( B \) is the inverse of \( A \). This property is foundational in solving equations where matrices are involved, making it a key concept in matrix algebra.

Matrix Algebra

Matrix algebra is a branch of mathematics dealing with the study of matrices and operations that involve them. It underpins numerous fields including physics, computer science, and engineering. Core operations include addition, subtraction, scalar multiplication, and matrix multiplication.

• The concept of an inverse matrix is central in matrix algebra because it allows for the solution of linear matrix equations.
• Not all matrices have inverses; a matrix must be square, and it must have a non-zero determinant to possess an inverse.
• In matrix algebra, solving equations with matrices often simplifies complex systems of linear equations.

In the context of this exercise, identifying the inverse matrix through algebraic manipulations helps solve matrix equations, and invertibility is confirmed when the matrix product yields the identity matrix. Explore these operations to enhance your understanding of matrix transformations and systems of equations.

Precalculus Mathematics

Precalculus mathematics often serves as the bridge between high school math and college-level calculus. It tackles several fundamental concepts, including the basics of matrices and their operations. Here, students are introduced to matrix algebra as a tool to solve complex mathematical problems efficiently.

• Understanding matrix operations is essential for tackling linear algebra problems, which are common in calculus and beyond.
• Precalculus provides the foundational knowledge of matrices, enabling students to grasp more advanced mathematical concepts later.
• Inverse matrices are a particular focus, as they have practical applications in solving systems of linear equations.

Through exercises like verifying the inverse of a matrix, students not only solidify their understanding of the topic but also prepare for future courses like calculus and linear algebra. Mastering these fundamentals is critical for progressing in mathematics and related fields.