Problem 27 - College Algebra

Question

In Exercises \(27-36,\) find (if possible): \(\begin{array}{llll}\text { a. } A B & \text { and } & \text { b. } B A\end{array}\) $$ A=\left[\begin{array}{ll} 1 & 3 \\ 5 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 3 & -2 \\ -1 & 6 \end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrices are AB = \[\left[\begin{array}{ll} 0 & 16 \ 12 & 8 \end{array}\right], and BA = \[\left[\begin{array}{ll} -7 & 3 \ 29 & 15 \end{array}\right]\]

Step 1: Check the dimensions for validity of multiplication

Matrix A is a \(2×2\) matrix and matrix B is also a \(2×2\) matrix. Hence, AB and BA are both valid as the number of columns in A equals the number of rows in B and vice versa.

Step 2: Calculate AB

Compute AB by multiplying corresponding elements and adding the values. Thus, AB equals \[\left[\begin{array}{ll} (1*3) + (3*-1) & (1*-2) + (3*6) \ (5*3) + (3*-1) & (5*-2) + (3*6) \end{array}\right] = \[\left[\begin{array}{ll} 0 & 16 \ 12 & 8 \end{array}\right]\]

Step 3: Calculate BA

Compute BA similarly. Thus, BA equals \[\left[\begin{array}{ll} (3*1) + (-2*5) & (3*3) + (-2*3) \ (-1*1) + (6*5) & (-1*3) + (6*3) \end{array}\right] = \[\left[\begin{array}{ll} -7 & 3 \ 29 & 15 \end{array}\right]\]

Key Concepts

Matrices

Matrices are rectangular arrays of numbers arranged in rows and columns. They serve as a foundational tool in linear algebra. Each element in a matrix is usually denoted by a coordinate that consists of its row and column position. In the given exercise, we work with two matrices:

• Matrix A, with elements aligned in two rows and two columns.
• Matrix B, which similarly has two rows and two columns.

The notation for a matrix can be compact, such as \[A = \begin{bmatrix} 1 & 3 \ 5 & 3 \end{bmatrix}\],where each element is identified by its specific row (horizontal) and column (vertical) place. Understanding matrices is the first step towards exploring complex structures and systems in various fields such as physics, computer science, and economics.

Matrix Operations

Matrix operations encompass several fundamental procedures such as addition, subtraction, and multiplication. Here, we focus on multiplication, which requires the matrices to have dimensions that align properly.
For matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. For this exercise, both matrices are 2x2, making them compatible for bending either as \(AB\) or \(BA\).When matrices are multiplied:

• Each element in the resulting product is obtained by summing the products of elements from the corresponding row and column.
• This can be computationally intensive but follows systematic rules, allowing precise computation.

Matrix multiplication is not commutative, meaning \(AB\) generally does not equal \(BA\). This is evident in the example, where different products emerge depending on the order of multiplication.

Algebraic Expressions

In working with matrices, algebraic expressions organize the calculations into manageable parts. An algebraic expression involves arithmetic calculations combining variables and numbers, set out in formulaic form.
For matrix multiplication, each element of the resulting matrix involves an algebraic expression defined by:

• The product of elements from a row of the first matrix with a column of the second matrix.
• All products from corresponding row and column elements are summed up.

For instance, when calculating the element at the first row and first column for \(AB\), the expression is \((1 \times 3) + (3 \times (-1)) = 0\).Handling these expressions correctly is crucial for ensuring that matrix computations are accurate and consistent.

2x2 Matrices

A 2x2 matrix is particularly simple yet forms the basis of understanding larger, more complex matrices. Each 2x2 matrix consists of four elements arranged in two rows and two columns.
In this exercise:

• Matrix A has elements \(\begin{bmatrix} 1 & 3 \ 5 & 3 \end{bmatrix}\).
• Matrix B has elements \(\begin{bmatrix} 3 & -2 \ -1 & 6 \end{bmatrix}\).

Multiplying two 2x2 matrices results in another 2x2 matrix. This property simplifies handling transformations and systems of equations efficiently.Recognizing how each element contributes during multiplication:

• Helps students appreciate how small systems are structured.
• Forms a stepping stone to comprehending larger dimension operations.

Mastering 2x2 matrices builds a strong foundation for advancing into more complex multivariable scenarios.