Math Problem Statement
A= ( 4 6 1 ) B= ( 2 4 6 ) ( 5 2 3 ) ( 1 8 2 ) ( 7 5 2 ) ( 3 7 1 ) tentukanlah: a. A+B b. A-B c. B-A d. A-(A+B) e. B+(A-B)
Solution
Given two 3x3 matrices and :
4 & 6 & 1 \\ 5 & 2 & 3 \\ 7 & 5 & 2 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 2 & 4 & 6 \\ 1 & 8 & 2 \\ 3 & 7 & 1 \end{pmatrix}$$ We will calculate the following operations: ### a. $$A + B$$ Matrix addition is done element-wise: $$A + B = \begin{pmatrix} 4+2 & 6+4 & 1+6 \\ 5+1 & 2+8 & 3+2 \\ 7+3 & 5+7 & 2+1 \end{pmatrix} = \begin{pmatrix} 6 & 10 & 7 \\ 6 & 10 & 5 \\ 10 & 12 & 3 \end{pmatrix}$$ ### b. $$A - B$$ Matrix subtraction is also done element-wise: $$A - B = \begin{pmatrix} 4-2 & 6-4 & 1-6 \\ 5-1 & 2-8 & 3-2 \\ 7-3 & 5-7 & 2-1 \end{pmatrix} = \begin{pmatrix} 2 & 2 & -5 \\ 4 & -6 & 1 \\ 4 & -2 & 1 \end{pmatrix}$$ ### c. $$B - A$$ This is the reverse of part b, calculated as: $$B - A = \begin{pmatrix} 2-4 & 4-6 & 6-1 \\ 1-5 & 8-2 & 2-3 \\ 3-7 & 7-5 & 1-2 \end{pmatrix} = \begin{pmatrix} -2 & -2 & 5 \\ -4 & 6 & -1 \\ -4 & 2 & -1 \end{pmatrix}$$ ### d. $$A - (A + B)$$ This expression simplifies as: $$A - (A + B) = A - A - B = -B$$ So: $$-B = \begin{pmatrix} -2 & -4 & -6 \\ -1 & -8 & -2 \\ -3 & -7 & -1 \end{pmatrix}$$ ### e. $$B + (A - B)$$ This can be simplified as: $$B + (A - B) = B + A - B = A$$ Thus: $$A = \begin{pmatrix} 4 & 6 & 1 \\ 5 & 2 & 3 \\ 7 & 5 & 2 \end{pmatrix}$$ ### Final Results: - **a.** $$A + B = \begin{pmatrix} 6 & 10 & 7 \\ 6 & 10 & 5 \\ 10 & 12 & 3 \end{pmatrix}$$ - **b.** $$A - B = \begin{pmatrix} 2 & 2 & -5 \\ 4 & -6 & 1 \\ 4 & -2 & 1 \end{pmatrix}$$ - **c.** $$B - A = \begin{pmatrix} -2 & -2 & 5 \\ -4 & 6 & -1 \\ -4 & 2 & -1 \end{pmatrix}$$ - **d.** $$A - (A + B) = \begin{pmatrix} -2 & -4 & -6 \\ -1 & -8 & -2 \\ -3 & -7 & -1 \end{pmatrix}$$ - **e.** $$B + (A - B) = A = \begin{pmatrix} 4 & 6 & 1 \\ 5 & 2 & 3 \\ 7 & 5 & 2 \end{pmatrix}$$ Would you like any further details on the calculations? ### 5 Questions: 1. How do you perform matrix multiplication? 2. What is the determinant of matrix $$A$$? 3. What is the inverse of matrix $$B$$, if it exists? 4. How can you find the transpose of a matrix? 5. How do you perform scalar multiplication on a matrix? ### Tip: Matrix operations like addition and subtraction are only possible for matrices of the same dimensions. Always check the size of matrices before performing these operations!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Operations
Matrix Addition
Matrix Subtraction
Formulas
A + B = [a_ij + b_ij] for all i,j
A - B = [a_ij - b_ij] for all i,j
B - A = [b_ij - a_ij] for all i,j
A - (A + B) = A - A - B = -B
B + (A - B) = B + A - B = A
Theorems
Element-wise matrix operations
Suitable Grade Level
Grades 10-12
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