Math Problem Statement

Find a. Upper A plus Upper B​, b. Upper A minus Upper B​, c. 2 Upper A​, and d. negative 5 Upper A minus 4 Upper B. Upper A equals left bracket Start 3 By 2 Matrix 1st Row 1st Column 2 2nd Column negative 6 2nd Row 1st Column negative 9 2nd Column 0 3rd Row 1st Column 8 2nd Column 9 EndMatrix right bracket ​, Upper B equals left bracket Start 3 By 2 Matrix 1st Row 1st Column 0 2nd Column negative 7 2nd Row 1st Column 8 2nd Column 0 3rd Row 1st Column 3 2nd Column negative 6 EndMatrix right bracket

Solution

We are tasked to compute the following matrix operations based on the given matrices AA and BB:

A=[269089],B=[078036].A = \begin{bmatrix} 2 & -6 \\ -9 & 0 \\ 8 & 9 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & -7 \\ 8 & 0 \\ 3 & -6 \end{bmatrix}.

Part (a): A+BA + B

To add two matrices, add the corresponding elements of AA and BB.

= \begin{bmatrix} 2+0 & -6+(-7) \\ -9+8 & 0+0 \\ 8+3 & 9+(-6) \end{bmatrix} = \begin{bmatrix} 2 & -13 \\ -1 & 0 \\ 11 & 3 \end{bmatrix}.$$ ### Part (b): $$A - B$$ To subtract two matrices, subtract the corresponding elements of $$B$$ from $$A$$. $$A - B = \begin{bmatrix} 2 & -6 \\ -9 & 0 \\ 8 & 9 \end{bmatrix} - \begin{bmatrix} 0 & -7 \\ 8 & 0 \\ 3 & -6 \end{bmatrix} = \begin{bmatrix} 2-0 & -6-(-7) \\ -9-8 & 0-0 \\ 8-3 & 9-(-6) \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ -17 & 0 \\ 5 & 15 \end{bmatrix}.$$ ### Part (c): $$2A$$ To scale a matrix by a scalar, multiply each element of the matrix by the scalar. $$2A = 2 \cdot \begin{bmatrix} 2 & -6 \\ -9 & 0 \\ 8 & 9 \end{bmatrix} = \begin{bmatrix} 2 \cdot 2 & 2 \cdot -6 \\ 2 \cdot -9 & 2 \cdot 0 \\ 2 \cdot 8 & 2 \cdot 9 \end{bmatrix} = \begin{bmatrix} 4 & -12 \\ -18 & 0 \\ 16 & 18 \end{bmatrix}.$$ ### Part (d): $$-5A - 4B$$ To compute $$-5A - 4B$$, first calculate $$-5A$$ and $$-4B$$, then subtract. $$-5A = -5 \cdot \begin{bmatrix} 2 & -6 \\ -9 & 0 \\ 8 & 9 \end{bmatrix} = \begin{bmatrix} -10 & 30 \\ 45 & 0 \\ -40 & -45 \end{bmatrix},$$ $$-4B = -4 \cdot \begin{bmatrix} 0 & -7 \\ 8 & 0 \\ 3 & -6 \end{bmatrix} = \begin{bmatrix} 0 & 28 \\ -32 & 0 \\ -12 & 24 \end{bmatrix}.$$ Now, compute $$-5A - 4B$$: $$-5A - 4B = \begin{bmatrix} -10 & 30 \\ 45 & 0 \\ -40 & -45 \end{bmatrix} - \begin{bmatrix} 0 & 28 \\ -32 & 0 \\ -12 & 24 \end{bmatrix} = \begin{bmatrix} -10-0 & 30-28 \\ 45-(-32) & 0-0 \\ -40-(-12) & -45-24 \end{bmatrix} = \begin{bmatrix} -10 & 2 \\ 77 & 0 \\ -28 & -69 \end{bmatrix}.$$ --- ### Final Answers: 1. $$A + B = \begin{bmatrix} 2 & -13 \\ -1 & 0 \\ 11 & 3 \end{bmatrix}$$, 2. $$A - B = \begin{bmatrix} 2 & 1 \\ -17 & 0 \\ 5 & 15 \end{bmatrix}$$, 3. $$2A = \begin{bmatrix} 4 & -12 \\ -18 & 0 \\ 16 & 18 \end{bmatrix}$$, 4. $$-5A - 4B = \begin{bmatrix} -10 & 2 \\ 77 & 0 \\ -28 & -69 \end{bmatrix}$$. Would you like further details on the steps? --- ### Related Questions: 1. How do you determine if two matrices can be added or subtracted? 2. What properties of scalar multiplication are important in matrix operations? 3. Can matrix multiplication be performed on these matrices $$A$$ and $$B$$? Why or why not? 4. How does the determinant or inverse of a matrix relate to scalar multiplication? 5. Can you find the transpose of each result in this problem? **Tip:** Always ensure that matrices have compatible dimensions before attempting operations like addition, subtraction, or multiplication!

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Math Problem Analysis

Mathematical Concepts

Matrix Addition
Matrix Subtraction
Scalar Multiplication of Matrices

Formulas

Matrix Addition: (A + B)[i][j] = A[i][j] + B[i][j]
Matrix Subtraction: (A - B)[i][j] = A[i][j] - B[i][j]
Scalar Multiplication: (cA)[i][j] = c * A[i][j]

Theorems

Basic Properties of Matrix Operations

Suitable Grade Level

Grades 10-12