Problem 26 - Advanced Engineering Mathematics

Question

Write the given sum as a single-column matrix. $$ \left(\begin{array}{r} 2 \\ 1 \\ -1 \end{array}\right)+5\left(\begin{array}{r} -1 \\ -1 \\ 3 \end{array}\right)-2\left(\begin{array}{r} 3 \\ 4 \\ -5 \end{array}\right) $$

Step-by-Step Solution

Answer

The sum is \(\left(\begin{array}{r} -9 \\ -12 \\ 24 \end{array}\right)\).

Step 1: Scalar Multiplication

Calculate the product of the scalars with the matrices: \(5\left(\begin{array}{r} -1 \ -1 \ 3 \end{array}\right)\) and \(-2\left(\begin{array}{r} 3 \ 4 \ -5 \end{array}\right)\).\This gives: \\(5\left(\begin{array}{r} -1 \ -1 \ 3 \end{array}\right) = \left(\begin{array}{r} -5 \ -5 \ 15 \end{array}\right)\) \and \\(-2\left(\begin{array}{r} 3 \ 4 \ -5 \end{array}\right) = \left(\begin{array}{r} -6 \ -8 \ 10 \end{array}\right)\).

Step 2: Matrix Addition

Now, add the resulting matrices: 1. Start with \(\left(\begin{array}{r} 2 \ 1 \ -1 \end{array}\right)\) and \(\left(\begin{array}{r} -5 \ -5 \ 15 \end{array}\right)\)2. Sum gives \(\left(\begin{array}{r} 2-5 \ 1-5 \ -1+15 \end{array}\right) = \left(\begin{array}{r} -3 \ -4 \ 14 \end{array}\right)\).3. Then, add \(\left(\begin{array}{r} -6 \ -8 \ 10 \end{array}\right)\) to the result from the previous step:4. Final sum gives \(\left(\begin{array}{r} -3-6 \ -4-8 \ 14+10 \end{array}\right) = \left(\begin{array}{r} -9 \ -12 \ 24 \end{array}\right)\).

Step 3: Conclusion

The given sum simplifies to the single-column matrix \(\left(\begin{array}{r} -9 \ -12 \ 24 \end{array}\right)\). This is the final result.

Key Concepts

Scalar Multiplication in Matrices

Scalar multiplication in matrices involves multiplying every element of a matrix by a scalar value. A scalar, simply put, is just a single number. This process is like distributing multiplication across all entries of the matrix. For example, if you have a scalar value of 5 and a matrix \( \left(\begin{array}{c} -1 \ -1 \ 3 \end{array}\right) \), then multiply each element within the matrix by 5.

• The first entry becomes \(-1 \times 5 = -5\).
• The second entry is also \(-1 \times 5 = -5\).
• The third entry becomes \(3 \times 5 = 15\).

The resulting matrix from multiplying \( \left(\begin{array}{c} -1 \ -1 \ 3 \end{array}\right) \) by 5 is \( \left(\begin{array}{c} -5 \ -5 \ 15 \end{array}\right) \). This process is similar regardless of the scalar or matrix involved.

Matrix Addition

Matrix addition involves adding two matrices of the same dimensions by adding their corresponding elements. To sum two matrices, each element in the first matrix pairs with the corresponding element in the second matrix. This operation can be visualized as stacking two matrices and adding vertically across corresponding entries.For instance, given matrices \( \left(\begin{array}{c} 2 \ 1 \ -1 \end{array}\right) \) and \( \left(\begin{array}{c} -5 \ -5 \ 15 \end{array}\right) \), their addition proceeds as follows:

• The first element becomes \(2 + (-5) = -3\).
• The second element is \(1 + (-5) = -4\).
• The third element becomes \(-1 + 15 = 14\).

The resulting matrix is \( \left(\begin{array}{c} -3 \ -4 \ 14 \end{array}\right) \). Similarly, continue this process for any subsequent matrices in your operation.

Single-Column Matrix

A single-column matrix, also known as a column vector, is a matrix with only one column and can have one or multiple rows. This type of matrix is often used to represent vectors in Cartesian coordinates, and it's very useful in various mathematical applications such as linear transformations and solving systems of equations.When dealing with matrix operations such as addition or scalar multiplication, the single-column matrix behaves similarly to larger matrices, adhering to the same rules. For instance, when performing addition, ensure that all matrices involved have the same number of rows. If you're adding two single-column matrices together, both should ideally have the same size.In the context of the exercise provided, the end result is a single-column matrix: \( \left(\begin{array}{c} -9 \ -12 \ 24 \end{array}\right) \). This demonstrates how transformations and operations can be expressed neatly with such matrices, making them a versatile and compact mathematical tool.