Problem 11 - College Algebra

Question

Find the products \(A B\) and \(B A\) to determine whether \(B\) is the multiplicative imerse 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 calculating the products \(AB\) and \(BA\), compare these products with the identity matrix to determine if \(B\) is the multiplicative inverse of \(A\).

Step 1: Calculate the product \(AB\)

To calculate the product \(AB\), take each row in \(A\) one-by-one and multiply it by each column in \(B\) in the same order. This will give you the matrix \(AB\).

Step 2: Calculate the product \(BA\)

By repeating the above process, compute the product \(BA\). This time, take each row in \(B\) one-by-one and multiply it by each column in \(A\) in the same order.

Step 3: Determine if \(B\) is a multiplicative inverse of \(A\)

To determine if \(B\) is the multiplicative inverse of \(A\), check if the computed products from Step 1 and 2 form an Identity matrix. If both \(AB\) and \(BA\) are the identity matrix, then \(B\) is the multiplicative inverse.

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential concept in algebra, often visualized as a way to apply a transformation represented by one matrix to another. It's not as straightforward as multiplying individual elements, however. To multiply two matrices, one must follow a specific procedure where you take the rows of the first matrix and columns of the second matrix to produce another matrix.

For instance, when you multiply a matrix by another, you calculate the entries of the resulting matrix by taking the dot product of the rows of the first matrix with the columns of the second. However, this can only be done if the number of columns in the first matrix matches the number of rows in the second.

In the context of our exercise, finding the product of matrices A and B involves this process. Keep in mind that matrix multiplication is not commutative - this means that generally, \(AB eq BA\). So, it's important to calculate both products individually to compare them.

Identity Matrix

An Identity matrix plays a role similar to the number one in matrix algebra. It's a special matrix that, when multiplied with another matrix, leaves the other matrix unchanged. In other words, it's the neutral element of matrix multiplication.

A key characteristic of an Identity matrix, denoted as \(I\), is that all of its diagonal entries (from the upper left corner to the bottom right corner) are one, and all off-diagonal entries are zero. They come in different sizes, but their structure remains the same regardless of size.

In the context of our textbook problem, we use an identity matrix as a reference point to determine if matrix B is indeed the multiplicative inverse of matrix A. If multiplying A by B (or B by A) yields an Identity matrix, then B qualifies as the multiplicative inverse of A.

Inverse of a Matrix

The inverse of a matrix, as its name implies, reverses the effect of applying the original matrix. For a square matrix A, the inverse is often denoted as \(A^{-1}\), and it has this special property: \(A \times A^{-1} = A^{-1} \times A = I\), where \(I\) is the identity matrix.

Finding the inverse of a matrix is not always possible. For a matrix to have an inverse, it must be square and have a non-zero determinant. In practice, computing the inverse involves more advanced techniques like row reduction or using the adjugate matrix. In the exercise, we're testing if B is the inverse of A by verifying if their product yields the identity matrix in both orders of multiplication.

Algebraic Operations on Matrices

Algebraic operations on matrices, such as addition, subtraction, and multiplication, follow specific rules that differ from simple arithmetic. When adding or subtracting matrices, they must be of the same size, and you add or subtract the corresponding elements. Multiplication, as covered earlier, is more complex - and division of matrices is not defined, but instead, we work with the concept of multiplicative inverse.

These operations allow us to solve systems of equations, transform geometrical figures, and work efficiently with multiple datasets in fields such as economics, engineering, and sciences. Each algebraic operation plays a unique role, and understanding the conditions and results of these operations is vital for anyone venturing into linear algebra or related fields.