Problem 10 - Precalculus with Limits

Question

In Exercises \(5-12,\) show that \(B\) is the inverse of \(A .\) $$A=\left[\begin{array}{rrr}{-4} & {1} & {5} \\ {-1} & {2} & {4} \\ {0} & {-1} & {-1}\end{array}\right], B=\left[\begin{array}{rrr}{-\frac{1}{2}} & {1} & {\frac{3}{2}} \\ {\frac{1}{4}} & {-1} & {-\frac{11}{4}} \\ {-\frac{1}{4}} & {1} & {\frac{7}{4}}\end{array}\right]$$

Step-by-Step Solution

Answer

Upon multiplying matrices A and B (both AB and BA), it can be seen that the resulting matrices are the identity matrix. Therefore, it can be concluded that B is indeed the inverse of A.

Step 1: Understand the concept of an inverse matrix

The inverse of a matrix A is another matrix, denoted as \(A^{-1}\), with the property that when they are multiplied together (in either order), the result is the identity matrix \(I\). In this case, to prove that B is the inverse of A, it needs to be shown that \(AB = BA = I\).

Step 2: Multiply matrix A by matrix B

To compute the product AB, perform the matrix multiplication method. It involves taking each row of the first matrix (A) and multiplying it element by element with each column of the second matrix (B), then adding the results. This new matrix is the result of the product AB.

Step 3: Check if AB equals the identity matrix

After finding the product AB, check if it equals the identity matrix. The identity matrix is a special matrix with ones on the diagonal and zeroes everywhere else. If AB equals the identity matrix, it means that B is indeed the inverse of A.

Step 4: Multiply matrix B by matrix A

Repeat the method for multiplication used in Step 2, but this time compute the product BA.

Step 5: Check if BA equals the identity matrix

After finding the product BA, check again if it equals the identity matrix. If it does, then it provides further solidification that B is the inverse of A.

Key Concepts

Matrix Multiplication

When learning about matrices in precalculus, one of the essential operations is matrix multiplication. Matrix multiplication is a bit different from multiplying numbers because you can't simply multiply the elements together. Instead, the process involves a specific set of steps.To multiply two matrices, such as matrix A and B, you need to calculate the dot product of rows from matrix A with columns from matrix B. For instance, to find the element in the first row and first column of the product matrix, you would multiply each element of the first row of A by the corresponding element of the first column of B, and then sum those products.

• The number of columns in matrix A must match the number of rows in matrix B to perform multiplication.
• The resulting matrix will have the same number of rows as matrix A and the same number of columns as matrix B.
• This process must be repeated for each row of matrix A and each column of matrix B.

The result of the multiplication will give you a new matrix where each element comes from these dot products.

Identity Matrix

In the context of linear algebra, the identity matrix plays a role equivalent to the number 1 in regular multiplication. It's a square matrix with ones on the main diagonal (from the top left to the bottom right) and zeroes in all other positions.Here's why the identity matrix is special:

• When any matrix A is multiplied by an identity matrix of the same size, the result is A.
• It serves as the multiplicative identity in the matrix world, meaning it does not change the matrix it multiplies.
• An identity matrix is denoted by a capital letter I, and its size is usually understood from the context.

When finding the inverse of a matrix, the goal is to identify a matrix which, when multiplied with the original matrix, results in the identity matrix. It confirms the property of an inverse precisely because it leaves the original matrix unchanged, akin to multiplying a number by 1.

Precalculus Foundations

Precalculus serves as the groundwork for understanding calculus and higher-level math. It includes topics such as functions, complex numbers, trigonometry, and matrices. A solid grasp of precalculus concepts is vital because it provides the necessary skills for calculating the inverse of a matrix, among other things.To succeed in exercises involving matrix inverses, students need to:

• Be comfortable with matrix operations like addition, multiplication, and finding determinants.
• Understand the properties of matrix inverses and the identity matrix.
• Apply logical steps to solve matrix equations, underscoring the importance of procedure in obtaining accurate results.

Most notably, exercises where the inverse of a matrix is to be demonstrated, like the one provided, allow students to use these foundational precalculus skills to explore and confirm the defining properties of matrix inverses. Through consistent practice and engagement with the step-by-step solutions, students can develop a deeper understanding and proficiency in these crucial mathematical concepts.