Math Problem Statement

Solve the given scalar matrix multiplication problems:

Solution

Let’s solve each matrix multiplication problem step by step:

1. 2(3−4)2 \begin{pmatrix} 3 \\ -4 \end{pmatrix}

We multiply each element of the matrix by 22:

\begin{pmatrix} 2 \cdot 3 \\ 2 \cdot -4 \end{pmatrix} = \begin{pmatrix} 6 \\ -8 \end{pmatrix}.$$ --- ### **2. $$-3 \begin{pmatrix} -2 \\ 12 \end{pmatrix}$$** We multiply each element of the matrix by $$-3$$: $$-3 \begin{pmatrix} -2 \\ 12 \end{pmatrix} = \begin{pmatrix} -3 \cdot -2 \\ -3 \cdot 12 \end{pmatrix} = \begin{pmatrix} 6 \\ -36 \end{pmatrix}.$$ --- ### **3. $$6 \begin{pmatrix} -1 & 4 \\ 3 & -2 \end{pmatrix}$$** We multiply each element of the matrix by $$6$$: $$6 \begin{pmatrix} -1 & 4 \\ 3 & -2 \end{pmatrix} = \begin{pmatrix} 6 \cdot -1 & 6 \cdot 4 \\ 6 \cdot 3 & 6 \cdot -2 \end{pmatrix} = \begin{pmatrix} -6 & 24 \\ 18 & -12 \end{pmatrix}.$$ --- ### **4. $$\frac{1}{3} \begin{pmatrix} -12 \\ 6 \end{pmatrix}$$** We multiply each element of the matrix by $$\frac{1}{3}$$ (or divide by 3): $$\frac{1}{3} \begin{pmatrix} -12 \\ 6 \end{pmatrix} = \begin{pmatrix} \frac{-12}{3} \\ \frac{6}{3} \end{pmatrix} = \begin{pmatrix} -4 \\ 2 \end{pmatrix}.$$ --- ### **5. $$\frac{1}{2} \begin{pmatrix} 4 & -10 \\ 2 & -18 \end{pmatrix}$$** We multiply each element of the matrix by $$\frac{1}{2}$$ (or divide by 2): $$\frac{1}{2} \begin{pmatrix} 4 & -10 \\ 2 & -18 \end{pmatrix} = \begin{pmatrix} \frac{4}{2} & \frac{-10}{2} \\ \frac{2}{2} & \frac{-18}{2} \end{pmatrix} = \begin{pmatrix} 2 & -5 \\ 1 & -9 \end{pmatrix}.$$ --- ### **6. $$\frac{1}{5} \begin{pmatrix} 25 & -30 \\ -100 & 35 \end{pmatrix}$$** We multiply each element of the matrix by $$\frac{1}{5}$$ (or divide by 5): $$\frac{1}{5} \begin{pmatrix} 25 & -30 \\ -100 & 35 \end{pmatrix} = \begin{pmatrix} \frac{25}{5} & \frac{-30}{5} \\ \frac{-100}{5} & \frac{35}{5} \end{pmatrix} = \begin{pmatrix} 5 & -6 \\ -20 & 7 \end{pmatrix}.$$ --- ### Final Answers: 1. $$\begin{pmatrix} 6 \\ -8 \end{pmatrix}$$ 2. $$\begin{pmatrix} 6 \\ -36 \end{pmatrix}$$ 3. $$\begin{pmatrix} -6 & 24 \\ 18 & -12 \end{pmatrix}$$ 4. $$\begin{pmatrix} -4 \\ 2 \end{pmatrix}$$ 5. $$\begin{pmatrix} 2 & -5 \\ 1 & -9 \end{pmatrix}$$ 6. $$\begin{pmatrix} 5 & -6 \\ -20 & 7 \end{pmatrix}$$ --- Do you need further clarification or a step-by-step breakdown for any part? Let me know! Here are 5 related questions: 1. What is scalar multiplication of a matrix? 2. How do you add two matrices together? 3. What happens when you multiply a matrix by 0? 4. What is the result of multiplying a 2x2 matrix by a scalar $$k$$? 5. How do you perform matrix multiplication for non-square matrices? **Tip**: Always distribute the scalar to *every* element in the matrix carefully to avoid errors.

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Math Problem Analysis

Mathematical Concepts

Matrix Scalar Multiplication
Matrix Operations

Formulas

Scalar Multiplication: k * A = k * a_ij for all elements a_ij in matrix A

Theorems

Properties of Scalar Multiplication in Matrices

Suitable Grade Level

Grades 9-10

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