Problem 4 - College Algebra

Question

Find the products \(A B\) and \(B A\) to determine whether \(B\) is the multiplicative imerse of \(A\). $$A=\left[\begin{array}{rr}-2 & 4 \\\1 & -2\end{array}\right], \quad B=\left[\begin{array}{rr}1 & 2 \\\\-1 & -2\end{array}\right]$$

Step-by-Step Solution

Answer

Matrix B is not the multiplicative inverse of matrix A as none of the products AB or BA resulted in an identity matrix.

Step 1: Matrix Multiplication A * B

The first step is to multiply matrix A by matrix B. Each element of the resultant matrix is calculated by taking the product of the corresponding row of the first matrix and column of the second, and summing them up. Therefore, for the matrix A * B, the elements will be calculated as follows: Element (1,1) = (-2)*1 + 4*(-1) = -6,Element (1,2) = (-2)*2 + 4*(-2) = -12,Element (2,1) = 1*1 +(-2)*(-1)=3, Element (2,2) = 1*2 + (-2)*(-2)=6.This results in the matrix AB = [ [-6, -12], [ 3, 6] ].

Step 2: Matrix Multiplication B * A

Using the same process, now multiply matrix B by matrix A:Element (1,1) = 1*(-2) + 2*1 = 0,Element (1,2) = 1*4 +2*(-2)=0,Element (2,1) = (-1)*(-2) +(-2)*1=0,Element (2,2) = (-1)*4 + (-2)*(-2)=0.This results in the matrix BA = [ [0, 0], [ 0, 0] ].

Step 3: Checking for Inverse

Neither of the results, AB or BA, are identity matrices. In fact, BA is a zero matrix. Therefore, we can conclude that B is not the multiplicative inverse of matrix A.

Key Concepts

Multiplicative Inverse

The multiplicative inverse of a matrix is a concept in linear algebra referring to a matrix that, when multiplied with the original matrix, yields the identity matrix. This property is similar to the inverse in basic arithmetic, such as with numbers, where multiplying a number by its inverse gives 1.
To understand this concept in matrices, consider two matrices, say, A and B. Matrix B is considered the multiplicative inverse of matrix A if the product of A and B (in both arrangements A * B and B * A) results in the identity matrix.

• If A * B = I and B * A = I, where I is the identity matrix of the same size as A and B, then B is the inverse of A.
• Not all matrices have a multiplicative inverse. Only square matrices (matrices with the same number of rows and columns) may have an inverse.

In the given exercise, calculating both products A * B and B * A does not yield an identity matrix, meaning B is not the multiplicative inverse of A.

Zero Matrix

A zero matrix is a unique type of matrix where all the elements are zeros. In mathematical notation, a zero matrix may be represented symbolically by 0 with appropriate dimensions specified. For example, a 2x2 zero matrix is written as:\[0 = \begin{bmatrix} 0 & 0 \0 & 0 \end{bmatrix}\]In the context of matrix multiplication, multiplying any matrix by a zero matrix results in a zero matrix.

• Zero matrices serve as absorbing elements for matrix multiplication. This means any matrix multiplied by a zero matrix results in a zero matrix, regardless of the order.
• If BA is a zero matrix in standard matrix multiplication, it indicates that B cannot be the multiplicative inverse of A, as it fails to produce an identity matrix.

In the given problem, when B was multiplied by A, the result was a zero matrix, which shows B is not the inverse of A.

Identity Matrix

The identity matrix serves as the equivalent of the number 1 in matrix multiplication. With matrices, an identity matrix is a square matrix where all the entries of the principal diagonal are ones, and all other entries are zeros. For instance, the identity matrix of size 2x2 is represented as:\[I = \begin{bmatrix} 1 & 0 \0 & 1 \end{bmatrix}\]The identity matrix is significant because multiplying any matrix by an identity matrix, regardless of the order, leaves the original matrix unchanged.

• If a matrix A, when multiplied by another matrix B (both ways, A * B and B * A), results in the identity matrix, then they are inverses of each other.
• The identity matrix is crucial in determining the invertibility of a matrix. If no arrangement of multiplication of two matrices results in an identity matrix, they are not inverses.

In the exercise provided, neither of the products, A * B or B * A, resulted in the identity matrix, confirming that B is not the inverse of A.