Problem 22 - College Algebra with Modeling and Visualization

Question

Evaluate the matrix expression. $$-3\left[\begin{array}{rr}3 & 8 \\\\-1 & -9\end{array}\right]+5\left[\begin{array}{rr}4 & -8 \\\1 & 6\end{array}\right]$$

Step-by-Step Solution

Answer

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

Step 1: Scalar Multiplication with Matrix A

The first step is to multiply each element of the matrix \( \begin{bmatrix} 3 & 8 \ -1 & -9 \end{bmatrix} \) by the scalar \(-3\). Perform the multiplication as follows:\[\begin{bmatrix} -3 \cdot 3 & -3 \cdot 8 \ -3 \cdot (-1) & -3 \cdot (-9) \end{bmatrix} = \begin{bmatrix} -9 & -24 \ 3 & 27 \end{bmatrix}\]

Step 2: Scalar Multiplication with Matrix B

Now, we will multiply each element of the matrix \( \begin{bmatrix} 4 & -8 \ 1 & 6 \end{bmatrix} \) by the scalar \(5\). Perform the multiplication:\[\begin{bmatrix} 5 \cdot 4 & 5 \cdot (-8) \ 5 \cdot 1 & 5 \cdot 6 \end{bmatrix} = \begin{bmatrix} 20 & -40 \ 5 & 30 \end{bmatrix}\]

Step 3: Adding the Resulting Matrices

Add the resulting matrices from Step 1 and Step 2 together by adding corresponding elements:\[\begin{bmatrix} -9 & -24 \ 3 & 27 \end{bmatrix} + \begin{bmatrix} 20 & -40 \ 5 & 30 \end{bmatrix} = \begin{bmatrix} (-9 + 20) & (-24 + (-40)) \ (3 + 5) & (27 + 30) \end{bmatrix} = \begin{bmatrix} 11 & -64 \ 8 & 57 \end{bmatrix}\]

Key Concepts

Scalar Multiplication

Scalar multiplication in matrix algebra involves multiplying every element of a matrix by a constant, known as a scalar. The result is a new matrix where each element is scaled by that constant. It’s a straightforward yet powerful operation used often in matrix calculations.
Let's take a closer look at how this works. Suppose we have a matrix:

• [3 8] [-1 -9]

If we multiply this matrix by the scalar details on the process...

• -3 × 3 = -9
• -3 × 8 = -24
• -3 × (-1) = 3
• -3 × (-9) = 27

This gives us the matrix

• [-9 -24] [3 27]

Remember that scalar multiplication scales each element of the matrix by the scalar value. It’s crucial not to confuse this process with matrix multiplication, which involves a set of different rules.

Matrix Addition

Matrix addition is an operation where we add two matrices by adding their corresponding elements. For two matrices to be added, they must be of the same dimension. This means each matrix must have the same number of rows and columns.
For instance, let's consider two matrices:

• [-9 -24] [3 27]
• [20 -40] [5 30]

In matrix addition, we add each element from the first matrix to the corresponding element in the second matrix:

• -9 + 20 = 11
• -24 + (-40) = -64
• 3 + 5 = 8
• 27 + 30 = 57

Thus, the resulting matrix is:

• [11 -64] [8 57]

Matrix addition is straightforward. Just ensure the matrices are compatible, meaning they are of the same size. If they are not, addition becomes impossible.

Matrix Operations

Matrix operations encompass a variety of processes for working with matrices, including but not limited to scalar multiplication, matrix addition, and matrix multiplication. Each operation is unique but forms the backbone of matrix algebra. - **Scalar Multiplication**: Involves multiplying each matrix element by a scalar. It's useful for resizing or scaling matrix values. - **Matrix Addition**: Sums corresponding elements of two matrices, requiring both matrices to have identical dimensions. - **Matrix Multiplication**: A more complex operation involving a sequence of steps. Requires the number of columns in the first matrix to match the number of rows in the second. Matrix operations are pivotal in various fields like computer graphics, engineering, physics, and more, allowing us to perform transformations, solve systems of equations, and represent linear transformations in a concise manner.
Learning these operations paves the way for understanding more advanced topics in mathematics and computer science. With consistent practice, these concepts will become intuitive and form a solid foundation for future studies.