Problem 12 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

If both matrix products A*B and B*A result in the identity matrix, then B is indeed the inverse of A. If not, B is not the inverse of A.

Step 1: Multiplication of Matrix A by Matrix B

Using the definition of matrix multiplication, compute the product A*B by multiplying each row of A by each column of B. This will result in a new 4x4 matrix.

Step 2: Verification of Identity Matrix

Check whether the resulting matrix from step 1 is an identity matrix (diagonal elements equal to 1, and all other elements equal to 0). If it does not match the identity matrix then stop and conclude that B is not the inverse of A.

Step 3: Multiplication of Matrix B by Matrix A

If step 2 passed, continue by calculating the matrix product B*A using the same method of row by column multiplication described in step 1.

Step 4: Final Verification

Finally, check whether the resulting matrix from step 3 is identical to the identity matrix. If it is, you can conclude that B is indeed the inverse of A.

Key Concepts

Matrix Multiplication

Matrix multiplication involves a systematic method where the elements of one matrix's rows are multiplied and summed with the corresponding elements of another matrix's columns.
This process is essential when working with matrix inverses to verify potential inverse pairs. To multiply a 4x4 matrix, follow these simple steps:

• Ensure both matrices are conformable, meaning the number of columns in the first equals the number of rows in the second.
• Multiply each element of the row from the first matrix by every element of the column from the second matrix.
• Sum these multiplied values to get an entry in the resulting matrix.

In practical terms, for matrix A and matrix B, you multiply each element of the first row of A with each element of the corresponding column in B.
Repeat for all rows and columns to get the full product matrix. If A and B are true inverses, their multiplication results in another special matrix called the identity matrix.

Identity Matrix

An identity matrix is a crucial concept in understanding matrix inverses. It acts as the multiplicative identity in matrix algebra, similar to how 1 is the multiplicative identity for real numbers.
When any matrix is multiplied by an identity matrix, it remains unchanged. The identity matrix has distinguishing features that make it easy to recognize:

• It is a square matrix.
• All the diagonal elements are 1.
• All off-diagonal elements are 0.

In a 4x4 identity matrix, it looks like this:\[I = \begin{bmatrix}1 & 0 & 0 & 0 \0 & 1 & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & 1 \end{bmatrix}\]When verifying if two matrices are inverses, you multiply them both ways: \( A \times B \) and \( B \times A \).
If both products result in an identity matrix, you've successfully proven the matrices are inverses of each other.

4x4 Matrices

Handling 4x4 matrices can initially seem daunting due to their size, but the process remains methodical. Key points to remember when working with 4x4 matrices include:

• They consist of 16 elements arranged in 4 rows and 4 columns.
• Operations like addition, subtraction, and scalar multiplication follow standard matrix rules.
• For multiplication, ensure that you can pair the matrix with another 4x4 or a compatible size.

4x4 matrices are often used in complex applications, such as solving systems of equations, transformations in computer graphics, or engineering computations.
Their structure allows them to store more data than lower dimension matrices, offering detailed representations of relationships or transformations.
A solid understanding of how to manipulate these matrices, through multiplication and finding inverses, provides a powerful tool for solving advanced mathematical problems. Knowing how to navigate their unique structure efficiently opens up many practical possibilities in applied mathematics and sciences.