Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Perform each operation if possible. $$3\left[\begin{array}{rrr}6 & -1 & 4 \\ 2 & 8 & -3 \\ -4 & 5 & 6\end{array}\right]+5\left[\begin{array}{rrr}-2 & -8 & -6 \\ 4 & 1 & 3 \\ 2 & -1 & 5\end{array}\right]$$

Step-by-Step Solution

Answer

The resulting matrix is \( \begin{bmatrix} 8 & -43 & -18 \\ 26 & 29 & 6 \\ -2 & 10 & 43 \end{bmatrix} \).

Step 1: Multiply the First Matrix by Scalar

Multiply each element of the first matrix by the scalar 3. The matrix is \( \begin{bmatrix} 6 & -1 & 4 \ 2 & 8 & -3 \ -4 & 5 & 6 \end{bmatrix} \). This gives: \[ 3 \times \begin{bmatrix} 6 & -1 & 4 \ 2 & 8 & -3 \ -4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} 18 & -3 & 12 \ 6 & 24 & -9 \ -12 & 15 & 18 \end{bmatrix} \]

Step 2: Multiply the Second Matrix by Scalar

Multiply each element of the second matrix by the scalar 5. The matrix is \( \begin{bmatrix} -2 & -8 & -6 \ 4 & 1 & 3 \ 2 & -1 & 5 \end{bmatrix} \). This gives:\[ 5 \times \begin{bmatrix} -2 & -8 & -6 \ 4 & 1 & 3 \ 2 & -1 & 5 \end{bmatrix} = \begin{bmatrix} -10 & -40 & -30 \ 20 & 5 & 15 \ 10 & -5 & 25 \end{bmatrix} \]

Step 3: Add the Two Matrices

Add the resulting matrices from steps 1 and 2 by adding their corresponding elements:\[ \begin{bmatrix} 18 & -3 & 12 \ 6 & 24 & -9 \ -12 & 15 & 18 \end{bmatrix} + \begin{bmatrix} -10 & -40 & -30 \ 20 & 5 & 15 \ 10 & -5 & 25 \end{bmatrix} = \begin{bmatrix} 8 & -43 & -18 \ 26 & 29 & 6 \ -2 & 10 & 43 \end{bmatrix} \]

Key Concepts

Scalar Multiplication

Scalar multiplication involves taking a matrix and multiplying every element within that matrix by a constant, called the scalar. This operation is straightforward and operates under the principle that you simply distribute the scalar across each element of the matrix.

This means each individual element of a matrix \( A \) is transformed as follows if you're multiplying by a scalar \( k \):

• \( k \times A[i,j] = k imes a_{i,j} \)

If you have a matrix like \[\begin{bmatrix} 6 & -1 & 4 \ 2 & 8 & -3 \ -4 & 5 & 6 \ \end{bmatrix}\]and multiply it by 3, each element is multiplied:

• \( 3 \times 6 = 18 \)
• \( 3 \times -1 = -3 \)
• \( 3 \times 4 = 12 \)
• ... and so on for all elements.

This results in a new matrix where each element is the product of the scalar and the corresponding element of the original matrix. Scalar multiplication is useful for scaling matrices uniformly, modifying their size or weight without changing their structure.

Matrix Addition

Matrix addition is the process of adding two matrices by adding their corresponding elements. To perform this operation, both matrices must be of the same dimension. For example, if you have two 3x3 matrices, you can add them together element-wise to produce another 3x3 matrix.

Here's a step-by-step breakdown:

• Align the matrices in such a way that each element in the first matrix can be paired with the corresponding element in the second matrix.
• Add each pair of corresponding elements to get the new element in the resulting matrix.

For instance, adding the matrices:\[\begin{bmatrix} 18 & -3 & 12 \ 6 & 24 & -9 \ -12 & 15 & 18 \end{bmatrix}+\begin{bmatrix} -10 & -40 & -30 \ 20 & 5 & 15 \ 10 & -5 & 25 \end{bmatrix}\]is done by adding each element:

• \( 18 + (-10) = 8 \)
• \( -3 + (-40) = -43 \)
• \( 12 + (-30) = -18 \)
• ...this pattern continues for all elements.

The result of this operation is another matrix of the same dimensions, where each element reflects the sum of its corresponding elements in the input matrices. Matrix addition is commutative and associative, making it a foundational tool in matrix arithmetic.

Matrix Arithmetic

Matrix arithmetic is a broad term that encompasses several operations relevant to matrices, including scalar multiplication and addition, as we've seen, as well as subtraction and more complex operations like multiplication (not to be confused with scalar multiplication).

Here are a few principles:

• **Commutative properties**: Matrix addition is commutative, meaning \( A + B = B + A \).
• **Associative properties**: Both matrix addition and multiplication are associative, which means \( (A + B) + C = A + (B + C) \) and \( (AB)C = A(BC) \).
• **Distributive property**: When a scalar multiplies a matrix sum, it distributes over each element: \( k(A + B) = kA + kB \).

Matrix multiplication (as opposed to scalar multiplication) is not commutative; this means that \( AB \) might not equal \( BA \). That said, scalar multiplication and addition are simpler and obey many familiar rules.

Understanding these operations allows for a broad range of applications, from solving linear equations to transforming geometric data, making them crucial tools in fields such as computer graphics, physics, engineering, and data science.