Math Problem Statement
If \[ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix} = \begin{bmatrix} 7 & 11 \\ k & 23 \end{bmatrix}, then find the value of k. (A) 17 (B) 19 (C) 21 (D) 23
Solution
We are given a matrix equation, where the product of two matrices is equal to another matrix:
1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix} = \begin{bmatrix} 7 & 11 \\ k & 23 \end{bmatrix}$$ We are tasked with finding the value of $$k$$. ### Step-by-Step Matrix Multiplication: Let’s denote the two matrices we are multiplying as $$A$$ and $$B$$, and the result matrix as $$C$$, such that: $$C = A \times B$$ The multiplication of two 2x2 matrices follows this rule: $$C = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} (ae + bg) & (af + bh) \\ (ce + dg) & (cf + dh) \end{bmatrix}$$ For the given matrices, we have: $$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix}$$ Performing the matrix multiplication: $$C = \begin{bmatrix} (1 \cdot 3 + 2 \cdot 5) & (1 \cdot 1 + 2 \cdot 2) \\ (3 \cdot 3 + 4 \cdot 5) & (3 \cdot 1 + 4 \cdot 2) \end{bmatrix} = \begin{bmatrix} (3 + 10) & (1 + 4) \\ (9 + 20) & (3 + 8) \end{bmatrix} = \begin{bmatrix} 13 & 5 \\ 29 & 11 \end{bmatrix}$$ We know the result matrix should be: $$\begin{bmatrix} 7 & 11 \\ k & 23 \end{bmatrix}$$ By comparing the corresponding elements of the two matrices, we have: $$k = 29$$ Therefore, the correct value of $$k$$ is **29**, but this does not match any of the given answer choices. There might be an error in the question or answer choices, as the calculated value of $$k$$ is correct based on the matrix multiplication. Would you like to discuss this further or explore a different approach? Here are 5 related questions: 1. What are the rules for matrix multiplication for general $$m \times n$$ and $$n \times p$$ matrices? 2. Can we perform matrix multiplication on non-square matrices? 3. How do you find the inverse of a 2x2 matrix? 4. What is the determinant of a 2x2 matrix and how is it related to matrix inversion? 5. How would the value of $$k$$ change if the second matrix were altered? **Tip:** When multiplying matrices, always ensure that the number of columns in the first matrix equals the number of rows in the second matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix multiplication
Linear algebra
Formulas
Matrix multiplication formula: \[ C = A \times B = \begin{bmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{bmatrix} \]
Theorems
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Suitable Grade Level
College level or advanced high school