Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Find, if possible, \(A B\) and \(B A\). $$ A=\left[\begin{array}{ll} 4 & 8 \end{array}\right], \quad B=\left[\begin{array}{r} -3 \\ 2 \end{array}\right] $$

Step-by-Step Solution

Answer

AB is \([4]\), and BA is \(\begin{bmatrix}-12 & -24 \\ 8 & 16\end{bmatrix}\).

Step 1: Determine Matrix Dimensions

Identify the dimensions of matrices \( A \) and \( B \). Matrix \( A \) is a 1x2 matrix, and matrix \( B \) is a 2x1 matrix.

Step 2: Check Multiplicability of AB

To multiply matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. \( A \) is 1x2 and \( B \) is 2x1, therefore \( A \times B \) is possible and will result in a 1x1 matrix.

Step 3: Calculate A times B (AB)

Multiply each element in the row of \( A \) by the corresponding element in the column of \( B \), and sum the products: \[ AB = \left[ 4 \times (-3) + 8 \times 2 \right] = \left[ -12 + 16 \right] = \left[ 4 \right]. \]

Step 4: Check Multiplicability of BA

For \( B \times A \), matrix \( B \) is 2x1 and matrix \( A \) is 1x2. This product is possible because the number of columns in \( B \) equals the number of rows in \( A \). The result will be a 2x2 matrix.

Step 5: Calculate B times A (BA)

Compute \( BA \) by multiplying the elements of \( B \)'s rows with the elements of \( A \)'s columns: \[ BA = \left[ \begin{array}{cc} (-3)\times4 & (-3)\times8 \ 2\times4 & 2\times8 \end{array} \right] = \left[ \begin{array}{cc} -12 & -24 \ 8 & 16 \end{array} \right]. \]

Key Concepts

Matrix Dimensions

Matrix dimensions are key to understanding how matrix multiplication works. The dimensions of a matrix are represented as "rows x columns." This tells us the structure of the matrix, helping us understand how matrices can interact through operations like multiplication.

For example, let's consider matrices from the exercise. Matrix \( A \) has dimensions of 1x2, meaning it has 1 row and 2 columns. On the other hand, Matrix \( B \) is 2x1, which means it has 2 rows and 1 column. Knowing these dimensions provides a roadmap to determine whether these matrices can be multiplied together and what the resultant dimensions will be.

Understanding matrix dimensions is important because matrix multiplication isn't like regular multiplication. When you multiply matrices, the number of columns in the first matrix must match the number of rows in the second matrix. If this condition is met, you can perform the multiplication. Otherwise, it's simply not possible.

Matrix Units

Matrix units refer to the individual components or "cells" of a matrix. Each of these units can be a number that plays a role in operations involving the matrix. These numbers are organized into rows and columns, which define the matrix's dimension and shape.

When you perform matrix multiplication, each unit or element inside the matrix matters. You must multiply corresponding units from the matrices involved. Each row unit from one matrix multiplies with each column unit from another matrix. Then sum these products to find the final result.

For instance, multiplying Matrix \( A \) and \( B \) involves the elements from these matrices. You multiply each element's numerical value according to its position and then sum them according to the defined rules to achieve a resultant matrix.

1x2 Matrix

A 1x2 matrix is a special type of matrix that has 1 row and 2 columns. This structure means we have a horizontal collection of two elements or units.

In the example given, Matrix \( A \) is a 1x2 matrix: \( \begin{bmatrix} 4 & 8 \end{bmatrix} \). This simple form makes it versatile in multiplying with other matrices, provided they have a matching dimension pattern. If you want to multiply this 1x2 matrix, you need another matrix whose dimensions fit the criteria.

Because matrix \( A \) only spreads across one row, to multiply it with another matrix, like matrix \( B \), that other matrix needs a column count equal to Matrix \( A \)'s row count, thus making pairing possible for multiplication.

2x1 Matrix

A 2x1 matrix is comprised of 2 rows and 1 column, giving it a vertical orientation. It is another example of a matrix that can interact with others during operations like matrix multiplication.

Matrix \( B \) from the example is a classic 2x1 matrix: \( \begin{bmatrix} -3 \ 2 \end{bmatrix} \). The orientation and dimensions mean that it can connect with matrices like Matrix \( A \) for multiplication. This is because \( B \)'s row count (2) aligns with \( A \)'s column count (2).

When multiplying the 2x1 matrix with another matrix, each unit in the rows of the 2x1 matrix is multiplied with corresponding units in column pairs. Understanding this structure helps in determining the resulting matrix's dimensions after multiplication.