Problem 16 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression. $$ -1\left[\begin{array}{rr} 4 & 11 \\ -2 & -1 \\ 9 & 3 \end{array}\right]+\frac{1}{6}\left(\left[\begin{array}{rr} -5 & -1 \\ 3 & 4 \\ 0 & 13 \end{array}\right]+\left[\begin{array}{rr} 7 & 5 \\ -9 & -1 \\ 6 & -1 \end{array}\right]\right) $$

Step-by-Step Solution

Answer

The evaluated expression is\[\left[\begin{array}{rr} -\frac{11}{3} & -\frac{31}{3} \1 & \frac{3}{2} \ -8 & -1 \end{array}\right]\]

Step 1: Scalar Multiplication

Multiply the given scalar with each matrix element. The scalars are -1 and \(\frac{1}{6}\) which are multiplied with the respective matrices. The resulting matrices will be:\[-\left[\begin{array}{rr} 4 & 11 \ -2 & -1 \ 9 & 3 \end{array}\right] = \left[\begin{array}{rr} -4 & -11 \ 2 & 1 \ -9 & -3 \end{array}\right]\]and \[\frac{1}{6} \left(\left[\begin{array}{rr} -5 & -1 \ 3 & 4 \ 0 & 13 \end{array}\right] + \left[\begin{array}{rr} 7 & 5 \ -9 & -1 \ 6 & -1 \end{array}\right]\right)\]First we add the two matrices giving\[\left[\begin{array}{rr} 2 & 4 \ -6 & 3 \ 6 & 12 \end{array}\right]\]and then multiply each element with \(\frac {1}{6}\) giving \[\left[\begin{array}{rr} \frac{1}{3} & \frac{2}{3} \ -1 & \frac{1}{2} \ 1 & 2 \end{array}\right]\]

Step 2: Addition of Matrices

Add the two resultant matrices from step 1 element by element. The result of the addition is:\[\left[\begin{array}{rr} -4 & -11 \ 2 & 1 \ -9 & -3 \end{array}\right] + \left[\begin{array}{rr} \frac{1}{3} & \frac{2}{3} \ -1 & \frac{1}{2} \ 1 & 2 \end{array}\right] =\left[\begin{array}{rr} -4+\frac{1}{3} & -11+\frac{2}{3} \ 2-1 & 1+\frac{1}{2} \ -9+1 & -3+2 \end{array}\right] =\left[\begin{array}{rr} -\frac{11}{3} & -\frac{31}{3} \ 1 & \frac{3}{2} \ -8 & -1 \end{array}\right]\]

Key Concepts

Scalar Multiplication

Scalar multiplication in matrix arithmetic is a straightforward operation where a single number, known as the scalar, multiplies every element within the matrix. Imagine you have a matrix and you want to scale it uniformly. This is where scalar multiplication comes in handy.
For instance, consider a scalar \(-1\) multiplying the matrix \(\left[\begin{array}{rr} 4 & 11 \ -2 & -1 \ 9 & 3 \end{array}\right]\). By multiplying every element of this 3x2 matrix (3 rows and 2 columns) by \(-1\), you effectively reverse the sign of each element.
Here's the step-by-step arithmetic:

• Multiply \(4\) by \(-1\) to get \(-4\)
• Multiply \(11\) by \(-1\) to get \(-11\)
• Continue this pattern for the other elements: \(-2\) turns to \(2\), \(-1\) to \(1\), \(9\) to \(-9\), and \(3\) to \(-3\)

The resulting matrix after scalar multiplication is \(\left[\begin{array}{rr} -4 & -11 \ 2 & 1 \ -9 & -3 \end{array}\right]\). Scalar multiplication is beneficial for manipulating matrices comprehensively but simply by impacting every element equally.

Matrix Addition

Matrix addition is the operation where you add corresponding elements of two or more matrices, which must be of the same size. In the context of the given exercise, two matrices were first added before their result was multiplied by a scalar. When performing matrix addition, remember the following key points to ensure correctness:

• The matrices must be of the same dimensions (e.g., both 3x2 matrices) for the operation to be valid.
• Each element from the first matrix is directly added to the corresponding element in the second matrix.

In the exercise, after the scalar multiplication of other matrices, two matrices:\[\left[\begin{array}{rr} -5 & -1 \ 3 & 4 \ 0 & 13 \end{array}\right] \text{ and } \left[\begin{array}{rr} 7 & 5 \ -9 & -1 \ 6 & -1 \end{array}\right]\]were added element by element, resulting in:\[\left[\begin{array}{rr} 2 & 4 \ -6 & 3 \ 6 & 12 \end{array}\right]\]This matrix was then subject to scalar multiplication. Such operations are commonly used in applications like graphics processing, where multiple transformations are applied to the data in stages.

Matrix Operations

Matrix operations encompass a variety of actions or transformations performed on matrices, including scalar multiplication, addition, subtraction, and more. These operations are pivotal for dealing with systems of equations, transformations, and many real-world applications, such as computer graphics, physics simulations, and more advanced fields like machine learning.
In the presented exercise, the operations illustrated include scalar multiplication followed by matrix addition, which is a common sequence when simplifying or preparing matrices for further calculations. Here’s a quick recap of the operations in this scenario:

• The given matrices were first multiplied by their respective scalars.
• Subsequently, resultant matrices were added, by performing element-wise addition.
• The sequential application of these operations allows us to manipulate the matrices easily and integrate them into broader calculations.

By mastering matrix operations, one can greatly enhance their ability to solve complex mathematical problems and make sense of multi-dimensional data efficiently.