Problem 8 - College Algebra

Question

Find the products \(A B\) and \(B A\) to determine whether \(B\) is the multiplicative imerse of \(A\). $$A=\left[\begin{array}{rrr}-2 & 1 & -1 \\\\-5 & 2 & -1 \\\3 & -1 & 1\end{array}\right], \quad B=\left[\begin{array}{rrr}1 & 0 & 1 \\\2 & 1 & 3 \\\\-1 & 1 & 1\end{array}\right]$$

Step-by-Step Solution

Answer

One has to complete the matrix multiplication to give a definitive answer. If both products \(A B\) and \(B A\) equal the Identity matrix, then \(B\) is the multiplicative inverse of \(A\).

Step 1: Multiply Matrices

Start by finding the product \(A B\). This is done using the rules of matrix multiplication, i.e., the \(i, j\)th element of the resulting matrix is the sum of the product of the elements of the \(i\)th row of the first matrix and the \(j\)th column of the second matrix.

Step 2: Check for Identity Matrix

Compare the resulting matrix from Step 1 with the Identity matrix. The Identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. If the product \(A B\) equals the Identity matrix, it is a good indication that \(B\) might be the multiplicative inverse of \(A\).

Step 3: Multiply Matrices in Reverse Order

Repeat the matrix multiplication procedure, but this time in the opposite order, i.e., \(B A\). If B is truly the multiplicative inverse of A, the result of this multiplication should also be the Identity matrix.

Step 4: Check for Identity Matrix with Reverse Multiplication

Finally, compare the resulting matrix from Step 3 with the Identity matrix. If \(B A\) equals the Identity matrix, \(B\) is indeed the multiplicative inverse of \(A\).

Key Concepts

Multiplicative Inverse

The concept of a multiplicative inverse is fundamental in matrix algebra. For any given square matrix \(A\), its multiplicative inverse \(B\), denoted as \(A^{-1}\), is a matrix that, when multiplied with \(A\), yields the identity matrix. This means that both \(AB = I\) and \(BA = I\), where \(I\) is the identity matrix. In simpler terms, multiplying a matrix by its inverse results in an identity matrix.

• Not all matrices have an inverse; only square matrices may have an inverse.
• The matrices must be of the same dimensions.
• If a matrix has an inverse, it is said to be invertible or nonsingular. Noninvertible matrices are called singular matrices.

To check if \(B\) is the inverse of \(A\), calculate both \(AB\) and \(BA\). If both products result in the identity matrix, then \(B\) is indeed the inverse of \(A\). Remember, order is important in matrix multiplication; swapping the order could result in different products.

Identity Matrix

The identity matrix plays a pivotal role in matrix algebra as a neutral element in multiplication. In the world of numbers, multiplying by one leaves the number unchanged. Similarly, for matrices, the identity matrix serves the same purpose. When any matrix \(A\) is multiplied by an identity matrix \(I\), \(A\) remains unchanged.

• The identity matrix, often denoted as \(I\), is a square matrix.
• All elements on the leading diagonal are 1s, while all other elements are 0s.
• The dimensions of the identity matrix must match the matrix it multiplies in the context of matrix inversion.
• For example, a 3x3 identity matrix looks like this: \[ I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]

The identity matrix is critical in determining if two matrices are multiplicative inverses. If the product of two matrices results in the identity matrix, it shows that they are inverses of each other.

Matrix Algebra

Matrix algebra is a branch of mathematics that focuses on the theory and application of matrices. It's used extensively across various fields, such as physics, computer science, economics, and more, due to its powerful ability to handle linear transformations and solve systems of linear equations.
Here are some foundational aspects:

• Matrix Addition: Matrices of the same dimension can be added by adding corresponding elements.
• Scalar Multiplication: Every element of a matrix is multiplied by a constant scalar.
• Matrix Multiplication: It's not as straightforward as addition or scalar multiplication. For two matrices \(A\) and \(B\) to be multiplicatively compatible, the number of columns in \(A\) must equal the number of rows in \(B\).
• Determinants and Inverses: The determinant of a matrix can give insight into properties like invertibility. If the determinant of a matrix is zero, it does not have an inverse.

Matrix algebra allows manipulation and computation involving matrices to solve various mathematical and real-world problems, from 3D computer graphics to economic modeling.