Problem 4 - Algebra and Trigonometry

Question

Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A .\) \(A=\left[\begin{array}{rrr}{3} & {2} & {4} \\ {1} & {1} & {-6} \\ {2} & {1} & {12}\end{array}\right], \quad B=\left[\begin{array}{rrr}{9} & {-10} & {-8} \\\ {-12} & {14} & {11} \\ {-\frac{1}{2}} & {\frac{1}{2}} & {\frac{1}{2}}\end{array}\right]\)

Step-by-Step Solution

Answer

AB and BA both equal the identity matrix, so B is the inverse of A.

Step 1: Understand the Problem

You are asked to calculate the products of two matrices \( A \) and \( B \), specifically \( AB \) and \( BA \), to verify if \( B \) is the inverse of \( A \). This means both products should result in the identity matrix for \( B \) to be the inverse of \( A \).

Step 2: Multiply Matrices A and B

Perform matrix multiplication to find the product \( AB \). Use the formula for matrix multiplication: the element at row \( i \) and column \( j \) of the product matrix is computed as the dot product of the \( i \)-th row of the first matrix and the \( j \)-th column of the second matrix.

Step 2.1: Calculate \( AB[1,1] \)

Compute the first element of \( AB \): \( 3 \times 9 + 2 \times (-12) + 4 \times (-\frac{1}{2}) = 27 - 24 - 2 = 1 \).

Step 2.2: Calculate \( AB[1,2] \)

Compute the second element of \( AB \): \( 3 \times (-10) + 2 \times 14 + 4 \times \frac{1}{2} = -30 + 28 + 2 = 0 \).

Step 2.3: Calculate \( AB[1,3] \)

Compute the third element of \( AB \): \( 3 \times (-8) + 2 \times 11 + 4 \times \frac{1}{2} = -24 + 22 + 2 = 0 \).

Step 2.4: Continue Calculating Other Entries of AB

Follow the same steps to calculate all remaining elements of matrix \( AB \), so that the complete product is determined.

Step 3: Verify AB is the Identity Matrix

Check that your calculated \( AB \) is indeed the identity matrix \( \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \).

Step 4: Multiply Matrices B and A

Perform matrix multiplication to find the product \( BA \). Use the same process as in Step 2, but switch the order of multiplication to calculate \( BA \).

Step 5: Verify BA is the Identity Matrix

Check that your calculated \( BA \) is also the identity matrix \( \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \).

Step 10: Final Step: Confirm B is the Inverse of A

If both \( AB \) and \( BA \) are the identity matrix, then \( B \) is the inverse of \( A \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a way to combine matrices to form a new matrix. It is arguably one of the most essential operations in matrix algebra, used across many fields such as computer graphics, physics, and engineering. Unlike regular multiplication, matrix multiplication is not commutative, which means that the order matters—multiplying matrix A by B yields different results than multiplying B by A.
- To perform matrix multiplication, consider two matrices, A and B, where the number of columns in A equals the number of rows in B. - The entry in the resulting matrix at position (i, j) is calculated by taking the dot product of the i-th row of A and the j-th column of B.
Understanding this process is crucial. For example, if you multiply the first row of A by the first column of B, the result will occupy the first position of the new matrix, and so forth. This method allows for the transformation of data and is decisive when finding matrix inverses.

Identity Matrix

An identity matrix, often denoted as I, is a special kind of matrix that acts as the multiplicative identity in matrix algebra. Much like how multiplying a number by 1 leaves it unchanged, a matrix multiplied by an identity matrix will remain the same.
- An identity matrix is always a square matrix, meaning it has the same number of rows and columns.- It has 1s on the main diagonal (from the top left to the bottom right) and 0s elsewhere.
For instance, in a 3x3 matrix situation, the identity matrix is:\[\begin{bmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1\end{bmatrix}\]
The identity matrix is crucial when verifying the inverse of a matrix. If both products, AB and BA, result in this identity matrix, it confirms that B is indeed the inverse of A.

Matrix Algebra

Matrix algebra is the set of rules and methods used to perform operations on matrices. It includes processes like addition, multiplication, and finding inverses, serving as a backbone for solving linear equations and transformations in higher dimensions.
- In the context of this exercise, matrix algebra helps us understand how to compute the inverse of a matrix and why finding the matrix product results in the identity matrix. - It encompasses concepts like determinant, transpose, and inverse—all necessary for advanced computations.
One of these concepts, the inverse of a matrix, is pivotal. The inverse matrix of A, noted as A⁻¹, holds the property that when multiplied by A, yields the identity matrix I. Not all matrices have inverses; a matrix must be square and have a non-zero determinant to possess one.
Overall, matrix algebra provides the tools to represent and solve systems of linear equations and transform vectors in space, like translating, rotating, or scaling objects in graphical applications.