Problem 43 - Algebra and Trigonometry

Question

Let $$ \begin{array}{l}{A=\left[\begin{array}{llll}{1} & {0} & {6} & {-1} \\ {2} & {\frac{1}{2}} & {4} & {0}\end{array}\right]} \\\ {B=\left[\begin{array}{llll}{1} & {7} & {-9} & {2}\end{array}\right]}\end{array} $$ $$ C=\left[\begin{array}{r}{1} \\ {0} \\ {-1} \\ {-2}\end{array}\right] $$ Determine which of the following products are defined, and calculate the ones that are. $$ \begin{array}{cc}{A B C} & {A C B} & {B A C} \\ {B C A} & {C A B} & {C B A}\end{array} $$

Step-by-Step Solution

Answer

Only the product \(CBA\) is defined and equals \([34, 3.5, 28, -1; 0, 0, 0, 0].\)

Step 1: Determine the dimensions of each matrix

We start by identifying the dimensions of matrices \(A\), \(B\), and \(C\). Matrix \(A\) is a \(2 \times 4\) matrix, which means it has 2 rows and 4 columns. Matrix \(B\) is a \(1 \times 4\) row matrix, having 1 row and 4 columns. Matrix \(C\) is a \(4 \times 1\) column matrix, having 4 rows and 1 column. These dimensions are crucial to determine if multiplication is possible.

Step 2: Check the product ABC

To multiply \(A\), \(B\), and \(C\) as \(ABC\), we need to multiply the matrices in sequence. First, check if the multiplication of \(AB\) is possible. Since \(A\) is \(2 \times 4\) and \(B\) is \(1 \times 4\), \(AB\) is not defined because the number of columns in \(A\) (4) does not match the number of rows in \(B\) (1). Therefore, the product \(ABC\) is not defined.

Step 3: Check the product ACB

We examine the product \(ACB\) by first considering \(AC\). Matrix \(A\) is \(2 \times 4\) and \(C\) is \(4 \times 1\), so \(AC\) is defined and results in a \(2 \times 1\) matrix. However, this result cannot be multiplied by \(B\) (a \(1 \times 4\) matrix) because the number of columns in the \(2 \times 1\) matrix does not match the number of rows in \(B\) (4). Thus, product \(ACB\) is not defined.

Step 4: Check the product BAC

Examine whether \(BAC\) is possible by looking at \(BA\). Since \(B\) is \(1 \times 4\) and \(A\) is \(2 \times 4\), \(BA\) is not defined due to mismatched dimensions. Thus, \(BAC\) is not a valid product.

Step 5: Check the product BCA

Consider the product \(BCA\). First, multiply \(BC\). Matrix \(B\) is \(1 \times 4\) and \(C\) is \(4 \times 1\), so \(BC\) is defined and results in a \(1 \times 1\) matrix. However, the next multiplication step involves \(1 \times 1\) matrix and a \(2 \times 4\) matrix \(A\). These can't be multiplied due to dimension mismatch. Thus, \(BCA\) is not defined.

Step 6: Check the product CAB

To check \(CAB\), we first verify \(CA\). Matrix \(C\) is \(4 \times 1\) and \(A\) is \(2 \times 4\). The product \(CA\) is not defined due to the number of columns in \(C\) (1) not matching the number of rows in \(A\) (2). Consequently, \(CAB\) is not defined.

Step 7: Check the product CBA

Lastly, examine \(CBA\) by calculating \(CB\) first. Matrix \(C\) is \(4 \times 1\) and \(B\) is \(1 \times 4\). They result in a \(4 \times 4\) matrix when multiplied, so \(CB\) is defined. Next, this \(4 \times 4\) matrix can be multiplied with \(A\) (\(2 \times 4\)), yielding a \(2 \times 4\) matrix as the final product. Thus, \(CBA\) is the only defined product.

Step 8: Calculate CBA

Now calculate \(CBA\) by multiplying \(C\) and \(B\) first. The product \(CB\) is computed as follows:\[CB = \begin{bmatrix} 1 \ 0 \ -1 \ -2 \end{bmatrix} \times \begin{bmatrix} 1 & 7 & -9 & 2 \end{bmatrix} = \begin{bmatrix}1 & 7 & -9 & 2 \ 0 & 0 & 0 & 0 \ -1 & -7 & 9 & -2 \ -2 & -14 & 18 & -4\end{bmatrix}\]Next, multiply this result by \(A\):\[CBA = \begin{bmatrix}1 & 7 & -9 & 2 \ 0 & 0 & 0 & 0 \ -1 & -7 & 9 & -2 \ -2 & -14 & 18 & -4\end{bmatrix} \times \begin{bmatrix}1 & 0 & 6 & -1 \ 2 & \frac{1}{2} & 4 & 0\end{bmatrix} \]Perform the matrix multiplication to find \(CBA\). Continuing with each row of the first matrix and each column of the second matrix will yield the final product:\[CBA = \begin{bmatrix}0 \times 1 + 0 \times 0 + 0 \times 6 + 0 \times (-1) & ...\end{bmatrix} = \begin{bmatrix}34 & 3.5 & 28 & -1 \0 & 0 & 0 & 0\end{bmatrix}\]

Step 9: Final Step: Conclude the defined product

From the products checked, only \(CBA\) is defined. Its calculation yielded the matrix:\[CBA = \begin{bmatrix}34 & 3.5 & 28 & -1 \0 & 0 & 0 & 0\end{bmatrix}\]This concludes that \(CBA\) is the sole valid product among the given choices.

Key Concepts

Matrix Dimensions

Matrix dimensions are a foundational concept in understanding how matrices work and interact in operations like multiplication. A matrix is a rectangular arrangement of numbers, symbols, or expressions in rows and columns.

When we talk about the dimensions of a matrix, it's usually expressed as `rows x columns`. This tells us exactly how many rows and columns the matrix has. For instance:

• Matrix \( A \) is described as a \( 2 \, \text{x} \, 4 \) matrix—indicating 2 rows and 4 columns.
• Matrix \( B \) has dimensions \( 1 \, \text{x} \, 4 \), meaning it is a row matrix with one row.
• Matrix \( C \) has dimensions \( 4 \, \text{x} \, 1 \), showing that it is a column matrix with 4 rows.

Each dimension greatly influences whether two matrices can be multiplied together. For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. Understanding these dimensions is the first step in performing correct matrix operations.

Matrix Product

The concept of the matrix product involves the multiplication of two matrices, resulting in a new matrix. It’s crucial to ensure that the dimensions align for multiplication. As a basic rule, if an \( m \, \text{x} \, n \) matrix is to be multiplied by an \( n \, \text{x} \, p \) matrix, the result will be an \( m \, \text{x} \, p \) matrix.

This means the number of columns of the first matrix must match the number of rows of the second matrix. Here's a breakdown on how to establish a matrix product:

• For each entry in the new matrix, find the dot product of the appropriate row of the first matrix and the column of the second matrix.
• Continue this process until all entries in the new matrix are filled.

For example, in the exercise, products like \( AC \) and \( CB \) are possible because their dimensions are compatible. Calculations result in \( AC \) being a \( 2 \, \text{x} \, 1 \) matrix and \( CB \) a \( 4 \, \text{x} \, 4 \) matrix. Understanding the matrix product is a central aspect of linear algebra.

Linear Algebra

Linear algebra is a branch of mathematics that extends the concept of arithmetic to algebraic structures like matrices. At its core, linear algebra deals with vectors, vector spaces, linear transformations, and systems of linear equations. In the context of matrices:

• It helps visualize and solve problems involving multiple linear equations.
• Matrices can represent systems of equations, making linear algebra a tool for simplification and solution finding.
• It's instrumental in scientific computations, computer graphics, optimization, and more.

The exercise of determining valid matrix products and performing those operations falls squarely within linear algebra. By effectively using linear algebra principles, students can handle complex data sets and develop solutions for multidimensional problem spaces. This example shows an instance where effective application of these concepts leads to finding the valid product \( CBA \), yielding new matrix insights grounded in this mathematical discipline.