Problem 49 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

AB = \( \left[\begin{array}{cc} -3 & 1 \\ -4 & 6 \end{array}\right] \), BA = \( \left[\begin{array}{cc} 4 & 2 \\ 5 & -1 \end{array}\right] \).

Step 1: Determine the Multiplicability of AB

To determine if matrices \( A \) and \( B \) can be multiplied as \( AB \), we check the dimensions. Matrix \( A \) is a \( 2 \times 2 \) matrix, and matrix \( B \) is also a \( 2 \times 2 \) matrix. For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Since both are \( 2 \), matrix \( AB \) is possible and will be a \( 2 \times 2 \) matrix.

Step 2: Multiply A and B to Find AB

To find \( AB \), multiply the rows of \( A \) by the columns of \( B \), summing the products:\[AB = \left[\begin{array}{rr}1 \cdot (-2) + (-1) \cdot 1 & 1 \cdot 3 + (-1) \cdot 2 \2 \cdot (-2) + 0 \cdot 1 & 2 \cdot 3 + 0 \cdot 2 \\end{array}\right]\]Calculating each element, we get:- The first row, first column element: \( 1 \cdot (-2) + (-1) \cdot 1 = -2 - 1 = -3 \).- The first row, second column element: \( 1 \cdot 3 + (-1) \cdot 2 = 3 - 2 = 1 \).- The second row, first column element: \( 2 \cdot (-2) + 0 \cdot 1 = -4 + 0 = -4 \).- The second row, second column element: \( 2 \cdot 3 + 0 \cdot 2 = 6 + 0 = 6 \).Thus, \( AB = \left[\begin{array}{cc} -3 & 1 \ -4 & 6 \end{array}\right] \).

Step 3: Determine the Multiplicability of BA

To determine if matrices \( B \) and \( A \) can be multiplied as \( BA \), we again check dimensions. Both \( B \) and \( A \) are \( 2 \times 2 \) matrices. Thus the multiplication \( BA \) is also possible and will produce a \( 2 \times 2 \) matrix.

Step 4: Multiply B and A to Find BA

To find \( BA \), multiply the rows of \( B \) by the columns of \( A \), summing the products:\[BA = \left[\begin{array}{rr}-2 \cdot 1 + 3 \cdot 2 & -2 \cdot (-1) + 3 \cdot 0 \1 \cdot 1 + 2 \cdot 2 & 1 \cdot (-1) + 2 \cdot 0 \\end{array}\right]\]Calculating each element, we get:- The first row, first column element: \( -2 \cdot 1 + 3 \cdot 2 = -2 + 6 = 4 \).- The first row, second column element: \( -2 \cdot (-1) + 3 \cdot 0 = 2 + 0 = 2 \).- The second row, first column element: \( 1 \cdot 1 + 2 \cdot 2 = 1 + 4 = 5 \).- The second row, second column element: \( 1 \cdot (-1) + 2 \cdot 0 = -1 + 0 = -1 \).Thus, \( BA = \left[\begin{array}{cc} 4 & 2 \ 5 & -1 \end{array}\right] \).

Key Concepts

Matrix Dimensions

When dealing with matrices, understanding their dimensions is crucial. Matrix dimensions are represented in the format \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns.

• A matrix with 2 rows and 3 columns is a \( 2 \times 3 \) matrix.
• For square matrices, like \( 2 \times 2 \), the number of rows and columns are equal.

Before multiplying two matrices, check whether the number of columns in the first matrix equals the number of rows in the second. This means that for matrices \( A \) and \( B \), if \( A \) is a \( 2 \times 2 \) matrix and \( B \) is also a \( 2 \times 2 \) matrix, they can be multiplied. The resulting product matrix will also be \( 2 \times 2 \).
Exploring dimensions helps ensure it's possible to execute matrix multiplication and avoids errors in mathematical operations.

Matrix Operations

Matrix operations, like addition, subtraction, and multiplication, are fundamental in algebra.
For matrix multiplication:

• Each entry of the resulting matrix is computed by taking the dot product of corresponding rows and columns.
• Multiply each element of a row by the corresponding element of a column, and sum these products.

For example, to compute entry at first row, first column in \( AB \), multiply elements of the first row of \( A \) with those of the first column of \( B \), summing the result: \( 1 \cdot (-2) + (-1) \cdot 1 \).
The commutative property does not apply to matrix multiplication. Therefore, \( AB \) generally does not equal \( BA \), which is evident in exercises. For example, the products \( AB eq BA \) demonstrate this variance.Matrix operations provide a way to solve many algebraic problems and are used in diverse fields like physics and computer science.

Education in Algebra

Education in algebra forms the basis for understanding complex mathematical concepts. Matrix algebra is a part of this larger framework.

• It introduces students to multi-dimensional problem-solving.
• Matrix operations prepare students for advanced study in mathematics, engineering, and the sciences.

In learning matrix multiplication, students develop skills in orderly calculation, which are crucial steps towards more abstract mathematical thinking. \( AB \) and \( BA \) in matrix exercises highlight non-commutative properties, promoting critical thinking.
Algebra is foundational because it fosters analytical capabilities. These skills are not only vital in academic settings but are also applicable in everyday decision-making. Thus, education in algebra, including matrix operations, cultivates a deeper understanding of numerical patterns and relationships.