Problem 11 - College Algebra

Question

find the products \(A B\) and \(B A\) to determine whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{rrrr} 0 & 0 & -2 & 1 \\ -1 & 0 & 1 & 1 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & -1 \end{array}\right], \quad B=\left[\begin{array}{llll} 1 & 2 & 0 & 3 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 1 & 2 & 0 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

After following the above steps, it will become clear whether \(B\) is the multiplicative inverse of \(A\).

Step 1: Multiply Matrices A and B

To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In this case, both matrices \(A\) and \(B\) are 4x4, which means they can be multiplied. For the product of two matrices, each element at row \(i\) and column \(j\) of the new matrix is calculated as the sum of the product of corresponding elements from row \(i\) of matrix \(A\) and column \(j\) of matrix \(B\).

Step 2: Multiply Matrices B and A

Repeat the matrix multiplication process but this time with matrix \(B\) as the first matrix and matrix \(A\) as the second. Recalculate each element at row \(i\) and column \(j\) of the new matrix as the sum of the product of corresponding elements from row \(i\) of matrix \(B\) and column \(j\) of matrix \(A\).

Step 3: Check for multiplicative inverse

After calculating the two new matrices, check if they are identity matrices. The identity matrix comprising of ones down the diagonal and the rest of the elements as zeros. If either \(AB\) or \(BA\) does not result in an identity matrix or both, then Matrix \(B\) is not the multiplicative inverse of Matrix \(A\).

Key Concepts

Multiplicative Inverse

In the realm of linear algebra, the multiplicative inverse of a matrix plays an important role. Just like how a number has an inverse (e.g., the inverse of 3 is 1/3, since 3 × 1/3 = 1), a matrix can also have an inverse, but only under certain conditions.
For a matrix to have an inverse, it must be square (same number of rows and columns) and its determinant must not be zero. If a matrix A has an inverse, then their product is the identity matrix. This means both \(AB\) and \(BA\) should equal the identity matrix. If matrix \(B\) is the inverse of \(A\), their multiplication will yield the identity matrix.

Identity Matrix

The identity matrix, often denoted as \(I\), is a pivotal concept in matrix operations. It acts as the multiplicative identity for matrices.
If a matrix has a multiplicative inverse, multiplying it by its inverse results in the identity matrix, effectively acting as the 'number 1' in matrix math.
For instance, in a 4x4 matrix, the identity matrix looks like this:

• First row: 1, 0, 0, 0
• Second row: 0, 1, 0, 0
• Third row: 0, 0, 1, 0
• Fourth row: 0, 0, 0, 1

It features 1s along the diagonal and 0s elsewhere. So, if matrix products \(AB\) or \(BA\) produce this arrangement, it confirms their inverses.

4x4 Matrices

Let's dive into 4x4 matrices. These matrices are part of a family called square matrices, which have equal numbers of rows and columns.
4x4 matrices are larger than their more frequently encountered 2x2 or 3x3 counterparts, but they follow the same principles for operations such as addition, subtraction, and multiplication.
This can get complex quickly because the calculations to determine products or inverses involve many more steps and numbers.

• Size and Complexity: Make sure to count correctly when matching elements for multiplication.
• Applications: They are common in computer graphics, simulations, and environments where transformations are key.

Understanding how they work is fundamental when dealing with transformations in higher dimensions.

Matrix Product Calculation

Matrix product calculation is essential for finding if one matrix is the inverse of another.
It involves a methodical process, especially with large matrices like 4x4. Here's how you calculate the product of matrices:

• First, verify matrix dimensions to ensure the number of columns in the first matches the rows in the second.
• For each element in the result matrix, sum the products of corresponding elements from the relevant row and column of the given matrices.
• It's crucial to stay organized and account for each number systematically, often using a grid or table to keep track.

When you multiply matrices, the goal is often to see if the resulting matrix matches the identity matrix, thereby revealing if one matrix is the inverse of the other.