Problem 10 - College Algebra

Question

find the products \(A B\) and \(B A\) to determine whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{lll} 0 & 2 & 0 \\ 3 & 3 & 2 \\ 2 & 5 & 1 \end{array}\right] \quad B=\left[\begin{array}{rrr} -3.5 & -1 & 2 \\ 0.5 & 0 & 0 \\ 4.5 & 2 & -3 \end{array}\right] $$

Step-by-Step Solution

Answer

Yes, B is indeed the multiplicative inverse of A, because the products AB and BA both result in an Identity Matrix.

Step 1: Calculate A times B

Matrix multiplication is done element by element, where each element in the resulting matrix is the sum of the products of elements in the corresponding row of the first matrix and column of the second.\n\n For instance, to get the element at the first row and first column of the resultant, you would multiply the elements of the first row of A with the elements of the first column of B and sum them up. \n\nPerform this for all the elements of both matrices to get \(AB = \begin{bmatrix}1 & 0 & 0\ 0 & 1 & 0\ 0 & 0 & 1\end{bmatrix}\)

Step 2: Calculate B times A

Similarly, multiply matrix B with matrix A, again doing so element by element. Remember that matrix multiplication is not commutative, meaning BA is not necessarily equal to AB. \n\nSo calculate the elements resulting from B times A as done for A times B. \n\nThis gives \(BA = \begin{bmatrix}1 & 0 & 0\ 0 & 1 & 0\ 0 & 0 & 1\end{bmatrix}\)

Step 3: Compare the products with the identity matrix

The identity matrix \(I\) is a special square matrix with ones on the main diagonal and zeros everywhere else. Both \(AB\) and \(BA\) are equal to the identity matrix which means thatB is indeed the multiplicative inverse of A.

Key Concepts

Multiplicative Inverse

The concept of a multiplicative inverse in matrix algebra is fundamental for understanding how matrix division works. A matrix has a multiplicative inverse, denoted as \(B\), with another matrix \(A\) if their product yields the identity matrix \(I\). In simpler terms, when two matrices are multiplied, they give a result that is equivalent to the identity matrix:

• For a matrix \(A\), another matrix \(B\) is its multiplicative inverse if both \(AB = I\) and \(BA = I\).
• The identity matrix, \(I\), is a square matrix where all the elements of the principal diagonal are ones and all other elements are zeros.
• Not all matrices have a multiplicative inverse; only non-singular matrices (those with a non-zero determinant) have inverses.

In the provided exercise, both \(AB\) and \(BA\) resulted in the identity matrix, indicating that \(B\) is indeed the multiplicative inverse of \(A\). This symmetry shows that each can effectively "undo" the operations of the other, much like how division works with numbers.

Identity Matrix

The identity matrix, often represented as \(I\), plays an essential role in matrix algebra. It acts much like the number 1 in regular arithmetic because multiplying any matrix by \(I\) leaves it unchanged:

• An \(n \times n\) identity matrix will have "1"s along the diagonal from the top left to the bottom right, and "0"s elsewhere.
• When a matrix \(A\) is multiplied by the identity matrix — whether on the left or the right — the result is matrix \(A\) itself: \(IA = A\) and \(AI = A\).
• The identity matrix is critical when discussing inverses. If the product of two matrices is the identity matrix, it indicates that one is the inverse of the other.

Understanding and recognizing the identity matrix helps us determine if two matrices are inverses. In the exercise, it helps conclude that \(AB = I\) and \(BA = I\), supporting the fact that \(B\) is the inverse of \(A\).

Matrix Algebra

Matrix algebra involves performing arithmetic operations such as addition, subtraction, and multiplication with matrices. It extends these familiar concepts to a grid of elements, introducing new rules and properties specific to matrices:

• Addition and subtraction: Matrices must be of the same dimensions to be added or subtracted, and operations are done element-wise.
• Multiplication: A key operation that involves taking the dot product of rows and columns. It's crucial to note that matrix multiplication is not commutative, meaning \(AB\) may not equal \(BA\).
• Determinant: A valuable number that can indicate if a matrix has an inverse. A non-zero determinant means the matrix is invertible.
• Inverse: The inverse of a matrix \(A\) is another matrix \(B\) such that \(AB = BA = I\). Inverses are analogous to division in regular arithmetic.

The essence of matrix algebra is using matrices in real-world applications like systems of equations, graphics transformations, and more. A solid grasp of matrix concepts, especially multiplication, identity matrices, and determinants, empowers students to solve complex algebraic problems effectively.