Math Problem Statement
Find the following matrix product, if it exists. left bracket Start 3 By 3 Matrix 1st Row 1st Column 1 2nd Column 0 3rd Column 0 2nd Row 1st Column 0 2nd Column 1 3rd Column 0 3rd Row 1st Column 0 2nd Column 0 3rd Column 1 EndMatrix right bracket 1 0 0 0 1 0 0 0 1 left bracket Start 3 By 3 Matrix 1st Row 1st Column negative 3 2nd Column 5 3rd Column negative 9 2nd Row 1st Column negative 5 2nd Column negative 8 3rd Column negative 7 3rd Row 1st Column 0 2nd Column negative 1 3rd Column 2 EndMatrix right bracket −3 5 −9 −5 −8 −7 0 −1 2
Solution
To find the matrix product, we are given two matrices:
1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} -3 & 5 & -9 \\ -5 & -8 & -7 \\ 0 & -1 & 2 \end{bmatrix}$$ The matrix $$A$$ is the identity matrix, which means when we multiply it with any other matrix, the result will be the second matrix itself. This is because the identity matrix leaves the other matrix unchanged during multiplication. Thus, the product $$A \times B$$ is: $$A \times B = \begin{bmatrix} -3 & 5 & -9 \\ -5 & -8 & -7 \\ 0 & -1 & 2 \end{bmatrix}$$ So, the matrix product is: $$\begin{bmatrix} -3 & 5 & -9 \\ -5 & -8 & -7 \\ 0 & -1 & 2 \end{bmatrix}$$ This is the result. Would you like further details or have any questions? Here are 5 related questions to expand your understanding: 1. How do you multiply two general $$3 \times 3$$ matrices? 2. What is the role of the identity matrix in matrix multiplication? 3. How does matrix multiplication differ from scalar multiplication? 4. Can matrix multiplication be commutative? 5. What are the properties of the identity matrix? **Tip**: Always remember that multiplying any matrix by the identity matrix will leave the original matrix unchanged.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Multiplication
Identity Matrix
Formulas
Matrix multiplication: (AB)_{ij} = Σ(A_{ik} * B_{kj}) for all k
Properties of the identity matrix: I * A = A
Theorems
Identity Matrix Theorem
Suitable Grade Level
Grades 9-12