Problem 28 - A Graphical Approach to College Algebra

Question

Perform each operation if possible. $$-3\left[\begin{array}{rr} 3 & 8 \\ -1 & -9 \end{array}\right]+5\left[\begin{array}{rr} 4 & -8 \\ 1 & 6 \end{array}\right]$$

Step-by-Step Solution

Answer

The result is the matrix \(\begin{bmatrix} 11 & -64 \\ 8 & 57 \end{bmatrix}\).

Step 1: Distribute the Scalar

First, distribute the scalar of \(-3\) across each element of the first matrix. This involves multiplying each element of the matrix by \(-3\).For the matrix \(\begin{bmatrix} 3 & 8 \ -1 & -9 \end{bmatrix}\):\[-3 \cdot \begin{bmatrix} 3 & 8 \ -1 & -9 \end{bmatrix} = \begin{bmatrix} -3 \times 3 & -3 \times 8 \ -3 \times (-1) & -3 \times (-9) \end{bmatrix} = \begin{bmatrix} -9 & -24 \ 3 & 27 \end{bmatrix}\]

Step 2: Distribute the Scalar

Now distribute the scalar of \(5\) across each element of the second matrix. Multiply each element by \(5\).For the matrix \(\begin{bmatrix} 4 & -8 \ 1 & 6 \end{bmatrix}\):\[5 \cdot \begin{bmatrix} 4 & -8 \ 1 & 6 \end{bmatrix} = \begin{bmatrix} 5 \times 4 & 5 \times (-8) \ 5 \times 1 & 5 \times 6 \end{bmatrix} = \begin{bmatrix} 20 & -40 \ 5 & 30 \end{bmatrix}\]

Step 3: Add the Resulting Matrices

Now add the two resulting matrices element-wise:- For the top left element: \(-9 + 20 = 11\)- For the top right element: \(-24 + (-40) = -64\)- For the bottom left element: \(3 + 5 = 8\)- For the bottom right element: \(27 + 30 = 57\)So, the resulting matrix is:\[\begin{bmatrix} 11 & -64 \ 8 & 57 \end{bmatrix}\]

Key Concepts

Scalar Multiplication

Scalar multiplication in matrices is about multiplying every element of a matrix by a constant number known as a scalar. This operation stretches or shrinks the matrix by the scalar's factor. Imagine you have a matrix like \( \begin{bmatrix} 3 & 8 \ -1 & -9 \end{bmatrix} \). If you multiply this matrix by \(-3\), each element in the matrix gets multiplied by \(-3\):

\[-3 \cdot \begin{bmatrix} 3 & 8 \ -1 & -9 \end{bmatrix} = \begin{bmatrix} -3 \times 3 & -3 \times 8 \ -3 \times (-1) & -3 \times (-9) \end{bmatrix} = \begin{bmatrix} -9 & -24 \ 3 & 27 \end{bmatrix}\]

This simple yet powerful operation adjusts each matrix element and is a foundational tool in linear algebra. Remember, scalar multiplication preserves the structure of the matrix while altering its magnitude.

Matrix Addition

Matrix addition is an operation where two matrices of the same size are added together element by element. It's like adding numbers in a corresponding position in each matrix. For example, after performing scalar multiplication on two matrices, you can add them like this:

Let's say you have:
\[\begin{bmatrix} -9 & -24 \ 3 & 27 \end{bmatrix}\]and
\[\begin{bmatrix} 20 & -40 \ 5 & 30 \end{bmatrix} \]

To add these, simply add each corresponding element:
- Top left: \(-9 + 20 = 11\)
- Top right: \(-24 + (-40) = -64\)
- Bottom left: \(3 + 5 = 8\)
- Bottom right: \(27 + 30 = 57\)

The result is:
\[\begin{bmatrix} 11 & -64 \ 8 & 57 \end{bmatrix}\]
This operation is straightforward but essential, ensuring each matrix element is perfectly aligned and adjusted.

Element-wise Operations

Element-wise operations in matrices refer to performing operations on corresponding elements of two matrices. These operations require two matrices to have the same dimensions. Element-wise operations include addition, subtraction, multiplication, or division of each pair of corresponding elements.

The main focus in our exercise was on addition. For instance, if we add matrices:
\[\begin{bmatrix} -9 & -24 \ 3 & 27 \end{bmatrix}\]and
\[\begin{bmatrix} 20 & -40 \ 5 & 30 \end{bmatrix} \]

we perform the addition as follows:

• Top left: \(-9 + 20 = 11\)
• Top right: \(-24 + (-40) = -64\)
• Bottom left: \(3 + 5 = 8\)
• Bottom right: \(27 + 30 = 57\)

Each addition is done separately for each element, resulting in a new matrix. Understanding element-wise operations is crucial not only in mathematics but also in programming where such operations are frequently used for array manipulations.