Problem 27 - Advanced Engineering Mathematics

Question

In Problems 25-28, write the given sum as a single-column matrix. $$ \left(\begin{array}{rr} 2 & -3 \\ 1 & 4 \end{array}\right)\left(\begin{array}{r} -2 \\ 5 \end{array}\right)-\left(\begin{array}{rr} -1 & 6 \\ -2 & 3 \end{array}\right)\left(\begin{array}{r} -7 \\ 2 \end{array}\right) $$

Step-by-Step Solution

Answer

The sum is \( \begin{pmatrix} -38 \\ -2 \end{pmatrix} \).

Step 1: Multiply the First Matrix and Vector

First, we need to multiply the matrix \( \begin{pmatrix} 2 & -3 \ 1 & 4 \end{pmatrix} \) by the vector \( \begin{pmatrix} -2 \ 5 \end{pmatrix} \). This involves computing the dot products for each row of the matrix with the vector. The first element of the resulting column matrix is: \[2(-2) + (-3)(5) = -4 -15 = -19\]The second element is: \[1(-2) + 4(5) = -2 + 20 = 18\]So the result of this multiplication is the column matrix: \[\begin{pmatrix} -19 \ 18 \end{pmatrix}\]

Step 2: Multiply the Second Matrix and Vector

Next, we multiply the matrix \( \begin{pmatrix} -1 & 6 \ -2 & 3 \end{pmatrix} \) by the vector \( \begin{pmatrix} -7 \ 2 \end{pmatrix} \). Similarly, compute the dot products for each row.The first element of the resulting matrix is: \[(-1)(-7) + 6(2) = 7 + 12 = 19\]The second element is: \[(-2)(-7) + 3(2) = 14 + 6 = 20\]Thus, the product is the column matrix: \[\begin{pmatrix} 19 \ 20 \end{pmatrix}\]

Step 3: Subtract the Second Result from the First

Now, subtract the second resulting matrix \( \begin{pmatrix} 19 \ 20 \end{pmatrix} \) from the first resulting matrix \( \begin{pmatrix} -19 \ 18 \end{pmatrix} \).Perform the subtraction between corresponding elements:First element: \[-19 - 19 = -38\]Second element: \[18 - 20 = -2\]So, the result is: \[\begin{pmatrix} -38 \ -2 \end{pmatrix}\]

Step 4: Express the Final Result as a Single-Column Matrix

The final result from the computation is already in the form of a single-column matrix:\[\begin{pmatrix} -38 \ -2 \end{pmatrix}\]This expression is the sum written as a single-column matrix.

Key Concepts

Matrix Multiplication

Matrix multiplication is a key operation in linear algebra, involving two matrices or a matrix and a vector. It plays an important role in various applications such as computer graphics, physics, and engineering. To successfully execute matrix multiplication, certain rules must be followed.
First, the number of columns in the first matrix must equal the number of rows in the second matrix. This means if you have a matrix with dimensions \( m \times n \), you can multiply it by another matrix with dimensions \( n \times p \). The resulting matrix will have dimensions \( m \times p \).When multiplying a matrix by a vector, each element of the resulting column is the dot product of a row from the matrix and the vector. The operation involves:

• Multiplying corresponding elements from a row of the matrix and the column vector.
• Adding those products together to get a single number, which becomes an element in the resulting column matrix.

Let's look at an example. In the multiplication of matrix \( \begin{pmatrix} 2 & -3 \ 1 & 4 \end{pmatrix} \) by vector \( \begin{pmatrix} -2 \ 5 \end{pmatrix} \), we perform the following calculations:

• The first element: \( 2 \times (-2) + (-3) \times 5 = -4 - 15 = -19 \)
• The second element: \( 1 \times (-2) + 4 \times 5 = -2 + 20 = 18 \)

This results in the column matrix \( \begin{pmatrix} -19 \ 18 \end{pmatrix} \).

Column Matrix

A column matrix is a special type of matrix that consists of a single column of elements, making it a vertical array. It is often used to represent vectors in mathematics, allowing for operations like addition, subtraction, and multiplication with other matrices.
A column matrix can be expressed as \( \begin{pmatrix} a \ b \ \ c \end{pmatrix} \), where \( a, b, \) and \( c \) are individual elements. Each element of a column matrix is usually a component of some vector space, typically corresponding to spatial dimensions or other quantifiable data.A column matrix is simple but powerful in computational processes:

• Its design allows it to easily interact with matrices, specifically regarding transformations or translations in vector spaces.
• They serve as foundational elements in linear transformations and can represent forces, directions, and velocities in physics.

When multiplying a matrix by a column vector, as shown in the earlier multiplication operations, the outcome is another column matrix. For instance, multiplying matrix \( \begin{pmatrix} -1 & 6 \ -2 & 3 \end{pmatrix} \) by vector \( \begin{pmatrix} -7 \ 2 \end{pmatrix} \) results in the column matrix \( \begin{pmatrix} 19 \ 20 \end{pmatrix} \). The structure and simplicity of column matrices make them an integral part of linear algebra.

Vector Subtraction

Vector subtraction is crucial in analyzing differences between vectors, effectively "subtracting" one set of quantities from another. This operation is essential in physics and engineering for computing relative positions and changes. To subtract one vector from another, you subtract corresponding components directly.
Let's consider two vectors in the column matrix form: \( \begin{pmatrix} x_1 \ x_2 \end{pmatrix} \) and \( \begin{pmatrix} y_1 \ y_2 \end{pmatrix} \). The resulting vector is found by:

• Subtracting the first component \( y_1 \) from \( x_1 \).
• Subtracting the second component \( y_2 \) from \( x_2 \).

The outcome is a new column matrix: \( \begin{pmatrix} x_1 - y_1 \ x_2 - y_2 \end{pmatrix} \).In our exercise, vector subtraction was employed after completing the matrix multiplications:

• The first resulting vector: \( \begin{pmatrix} -19 \ 18 \end{pmatrix} \)
• The second resulting vector: \( \begin{pmatrix} 19 \ 20 \end{pmatrix} \)

We subtract the second vector from the first:

• First component: \( -19 - 19 = -38 \)
• Second component: \( 18 - 20 = -2 \)

This gives us the final result \( \begin{pmatrix} -38 \ -2 \end{pmatrix} \), demonstrating how matrix multiplication results culminate in vector subtraction to provide precise outcomes.