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
The given problem involves the multiplication of two matrices. Let's denote the two matrices as:
1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ and $$B = \begin{pmatrix} -3 & 5 & -9 \\ -5 & -8 & -7 \\ 0 & -1 & 2 \end{pmatrix}.$$ ### Step 1: Understand Matrix Multiplication Matrix multiplication is done by taking the dot product of the rows of the first matrix with the columns of the second matrix. ### Step 2: Perform the Multiplication Since $$A$$ is the identity matrix, the multiplication of $$A$$ with any matrix $$B$$ will yield $$B$$ itself. This is because the identity matrix has no effect on the result when multiplying it with any matrix. \[ A \cdot B = B = \begin{pmatrix} -3 & 5Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Multiplication
Identity Matrix
Formulas
Matrix product rule: C = A · B, where each element C_ij = sum of A_ik * B_kj for k = 1 to n
Properties of Identity Matrix
Theorems
Identity Matrix Theorem: Multiplying any matrix by an identity matrix results in the original matrix
Suitable Grade Level
Grades 11-12