Math Problem Statement
Given two matrices A and B, find the element C_1^2 of the matrix C = A · B. The upper index corresponds to the row number.
Solution
The problem involves matrix multiplication , where the matrices and are given. The task is to find the specific element of the resulting matrix . The element corresponds to the element in the first row and second column of the product .
Steps to Solve:
-
Matrix Multiplication: Calculate using the formula for the elements of the resulting matrix :
-
Focus on : To find , compute the dot product of the first row of with the second column of .
Matrices:
\begin{bmatrix} 3 & -3 & 1 & -3 \\ 4 & 0 & 5 & -1 \\ 4 & 0 & 4 & -2 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & -3 & 1 & -1 \\ 4 & -1 & 5 & 0 \\ 1 & -2 & 2 & 1 \\ 5 & 0 & 3 & -2 \end{bmatrix}.$$ ### First Row of $$A$$: $$[3, -3, 1, -3]$$ ### Second Column of $$B$$: $$[-3, -1, -2, 0]$$ ### Compute $$C_1^2$$: $$C_1^2 = 3 \cdot (-3) + (-3) \cdot (-1) + 1 \cdot (-2) + (-3) \cdot 0$$ $$C_1^2 = -9 + 3 - 2 + 0 = -8.$$ ### Final Answer: $$C_1^2 = -8.$$ Would you like a detailed breakdown of another part or clarification? --- ### Follow-Up Questions: 1. How is matrix multiplication defined for non-square matrices? 2. What are the dimensions of the resulting matrix when $$A$$ and $$B$$ are multiplied? 3. How would the computation change if the matrices were larger? 4. Can the process be simplified for sparse matrices? 5. How do the elements of $$A$$ influence the rows of $$C$$? ### Tip: When performing matrix multiplication, always ensure the **number of columns in the first matrix matches the number of rows in the second matrix**.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Linear Algebra
Formulas
C_ij = Σ(A_ik * B_kj), where k iterates over columns of A and rows of B
Theorems
Matrix Multiplication Rule
Suitable Grade Level
Undergraduate Mathematics or Advanced High School Algebra