Problem 3 - Advanced Engineering Mathematics

Question

In Problems 1-20, fill in the blanks or answer true/false. If \(\mathbf{A}=\left(\begin{array}{l}1 \\ 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{ll}3 & 4\end{array}\right)\), then \(\mathbf{A B}=\) _________ and \(\mathbf{B} \mathbf{A}=\)_________.

Step-by-Step Solution

Answer

\(\mathbf{A} \mathbf{B} = \begin{pmatrix} 3 & 4 \\ 6 & 8 \end{pmatrix}\); \(\mathbf{B} \mathbf{A} = \begin{pmatrix} 11 \end{pmatrix}\).

Step 1: Understand Matrix Dimensions

Matrix \(\mathbf{A}\) is a 2x1 matrix, which means it has 2 rows and 1 column. Matrix \(\mathbf{B}\) is a 1x2 matrix, meaning it has 1 row and 2 columns. Understanding dimensions is crucial for matrix multiplication.

Step 2: Check Compatibility for Multiplication

We can multiply two matrices if the number of columns in the first matrix equals the number of rows in the second matrix. For \(\mathbf{A} \mathbf{B}\), we have a 2x1 matrix and a 1x2 matrix. The first has 1 column, and the second has 1 row, so they are compatible. For \(\mathbf{B} \mathbf{A}\), we have a 1x2 matrix and a 2x1 matrix, also compatible due to the matching dimensions (2 and 2).

Step 3: Calculate \(\mathbf{A} \mathbf{B}\)

To find the product \(\mathbf{A} \mathbf{B}\), the resulting matrix will have dimensions 2x2 (from the outer dimensions of the original matrices). The entries will be calculated by taking the linear combination of the rows of \(\mathbf{A}\) and the columns of \(\mathbf{B}\). Specifically, \(\begin{pmatrix} 1 \times 3 & 1 \times 4 \ 2 \times 3 & 2 \times 4 \end{pmatrix} = \begin{pmatrix} 3 & 4 \ 6 & 8 \end{pmatrix}\).

Step 4: Calculate \(\mathbf{B} \mathbf{A}\)

To compute \(\mathbf{B} \mathbf{A}\), the resulting matrix is 1x1. This single entry is determined by multiplying the row of \(\mathbf{B}\) with the column of \(\mathbf{A}\): \(3 \times 1 + 4 \times 2 = 3 + 8 = 11\). So, \(\mathbf{B} \mathbf{A} = \begin{pmatrix} 11 \end{pmatrix}\).

Step 5: Fill in the Blanks

According to the calculations, \(\mathbf{A} \mathbf{B} = \begin{pmatrix} 3 & 4 \ 6 & 8 \end{pmatrix}\) and \(\mathbf{B} \mathbf{A} = \begin{pmatrix} 11 \end{pmatrix}\).

Key Concepts

Matrix Dimensions

When you're working with matrices, understanding their dimensions is your first key step. Dimensions of a matrix are expressed in the form of "rows x columns." In our exercise, matrix \( \mathbf{A} \) is a 2x1 matrix. This means it has:

• 2 rows
• 1 column

Matrix \( \mathbf{B} \) is a 1x2 matrix, which has:

• 1 row
• 2 columns

Grasping these dimensions is crucial because they determine how matrices can interact with each other.

Each element in the matrix is positioned by its row and column order, important for operations like multiplication. Thus, a good understanding of matrix dimensions sets a strong foundation for further calculations.

Matrix Compatibility

Before you can multiply matrices, you need to check if they are compatible. What this means is that the matrices must "agree" on certain dimensions. Specifically, matrix multiplication is only possible if the number of columns in the first matrix equals the number of rows in the second matrix.

Let's consider our matrices again:

• Matrix \( \mathbf{A} \) is 2x1, meaning it has 1 column.
• Matrix \( \mathbf{B} \) is 1x2, so it has 1 row.

The column count of \( \mathbf{A} \) matches the row count of \( \mathbf{B} \), so you can multiply \( \mathbf{A} \mathbf{B} \). Interestingly, you can also multiply \( \mathbf{B} \mathbf{A} \) because the column count of \( \mathbf{B} \) equals the row count of \( \mathbf{A} \), even though the resulting dimensions will be different.

This compatibility check is a vital step, ensuring that you're set up to perform the proper matrix operations. Skipping this check risks running into errors in calculation.

Matrix Product Calculation

Once you have dimensions and compatibility checked out, it's time to perform the actual multiplication. The resulting matrix from multiplying two matrices is determined by the outer dimensions of the original matrices.

For \( \mathbf{A} \mathbf{B} \):

• The dimensions will be 2x2 from our 2x1 and 1x2 matrices.
• The resulting matrix is formed by taking the linear combination of the rows of \( \mathbf{A} \) and columns of \( \mathbf{B} \):

\[ \mathbf{A} \mathbf{B} = \begin{pmatrix} 1 \times 3 & 1 \times 4 \ 2 \times 3 & 2 \times 4 \end{pmatrix} = \begin{pmatrix} 3 & 4 \ 6 & 8 \end{pmatrix} \]Similarly, for \( \mathbf{B} \mathbf{A} \):

• The resulting matrix is a 1x1 matrix, since it's 1 row by 1 column.
• This single-value matrix is a sum of products from the row of \( \mathbf{B} \) and column of \( \mathbf{A} \):

\[ \mathbf{B} \mathbf{A} = 3 \times 1 + 4 \times 2 = 11 \]These calculations show how matrix multiplication can vary in result depending on the order, illustrating the non-commutative nature of matrix multiplication. Taking the time to compute carefully will give you an accurate and meaningful product.

Always double-check your operations to ensure a precise matrix product.