Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Find, if possible, \(A B\) and \(B A\). $$ A=\left[\begin{array}{rrr} 2 & 0 & 1 \\ -1 & 2 & 0 \end{array}\right], \quad B=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 3 & 1 & 0 \\ 0 & 2 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

Only \( AB \) is possible and equals \( \begin{bmatrix} 2 & 0 & 5 \\ 5 & 3 & -2 \end{bmatrix} \). \( BA \) is not possible.

Step 1: Determine if Matrix Multiplication is Possible

Matrix multiplication is possible when the number of columns in the first matrix is equal to the number of rows in the second matrix. Matrix \( A \) is a \( 2 \times 3 \) matrix and matrix \( B \) is a \( 3 \times 3 \) matrix. Since the number of columns in \( A \) (3) is equal to the number of rows in \( B \) (3), \( AB \) can be computed. \( BA \), on the other hand, requires \( B \) to be \( 3 \times 3 \) and \( A \) to be \( 3 \times 2 \), making \( BA \) not possible here because \( B \)'s columns (3) and \( A \)'s rows (2) do not match.

Step 2: Multiply Matrices to Find \( AB \)

To find \( AB \), multiply each element of the rows of \( A \) with each element of the columns of \( B \) and sum them. Let \( C = AB \) be the resulting \( 2 \times 3 \) matrix. Compute element \( c_{11} = 2 \times 1 + 0 \times 3 + 1 \times 0 = 2 \), \( c_{12} = 2 \times -1 + 0 \times 1 + 1 \times 2 = 0 \), \( c_{13} = 2 \times 2 + 0 \times 0 + 1 \times 1 = 5 \). Compute \( c_{21} = -1 \times 1 + 2 \times 3 + 0 \times 0 = 5 \), \( c_{22} = -1 \times -1 + 2 \times 1 + 0 \times 2 = 3 \), and \( c_{23} = -1 \times 2 + 2 \times 0 + 0 \times 1 = -2 \). Thus, \( AB = \begin{bmatrix} 2 & 0 & 5 \ 5 & 3 & -2 \end{bmatrix} \).

Step 3: Confirm \( BA \) is Not Possible

Matrix multiplication \( BA \) requires the number of columns in matrix \( B \) to match the number of rows in matrix \( A \). Since \( B \) is a \( 3 \times 3 \) matrix and \( A \) is a \( 2 \times 3 \) matrix, the dimensions mismatch (3 columns in \( B \) and 2 rows in \( A \)), therefore \( BA \) cannot be computed.

Key Concepts

Matrix Dimensions

When we talk about matrices in mathematics, the concept of matrix dimensions is crucial. Each matrix has a specific size described in terms of rows and columns. For matrix operations like multiplication to work, understanding matrix dimensions is key.
Matrix dimensions are denoted as "rows by columns." This means a matrix with 2 rows and 3 columns is labeled as a \(2 \times 3\) matrix. In our original problem, matrix \(A\) is a \(2 \times 3\) matrix and matrix \(B\) is a \(3 \times 3\) matrix.

• The first number refers to the rows, and the second number refers to the columns.
• Dimensions must meet specific criteria for certain matrix operations.

Knowing these dimensions is essential before performing operations like multiplication, where the number of columns of the first matrix must equal the number of rows of the second matrix. If they do not match, the multiplication cannot be performed.

Matrix Operations

Matrix operations include various activities like addition, subtraction, and multiplication. Matrix multiplication is more complex than regular arithmetic multiplication. The main requirement for multiplying two matrices is that the number of columns in the first matrix must equal the number of rows in the second matrix.
In the exercise, multiplying \(A\) and \(B\), the number of columns in \(A\) corresponds with the number of rows in \(B\), allowing us to compute \(AB\). However, for \(BA\), the dimensions do not match because the columns of \(B\) do not match the rows of \(A\). As a result, \(BA\) is not computable.

• Matrix multiplication is not commutative, meaning \(AB\) is not necessarily equal to \(BA\).
• Each element in the resulting matrix is the dot product of a row from the first matrix and a column from the second matrix.

These rules make matrix operations diverse and flexible, but they do require understanding the underlying dimensional requirements.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vector spaces and linear mappings. Matrices are foundational elements in linear algebra, representing linear transformations and systems of linear equations.
Within linear algebra, understanding matrix multiplication is significant because it extends to various applications such as transformation of coordinate spaces, solving systems of equations, and even computer graphics.

• Matrices simplify complex linear transformations into calculable problems.
• They can represent numerous variables and equations compactly.

In many practical and theoretical fields, linear algebra and its concepts like matrix operations allow for elegant solutions to otherwise complex problems. Thus, mastery of these concepts opens doors to advanced mathematical applications.