Show by direct calculation for 2x2 matrices A, B, C that A(BC)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem: Matrix Associativity**
Show by direct calculation for 2×2 matrices \( A, B, C \) that \( A(BC) = (AB)C \).
To solve this problem, we'll need to verify the associative property of matrix multiplication for 2x2 matrices using direct calculation. Associative property in matrices means that the order in which the matrices are grouped does not affect the product.
1. **Define the matrices:**
- Let \( A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \),
- \( B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} \),
- \( C = \begin{bmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{bmatrix} \).
2. **Compute \( BC \):**
\[
BC = \begin{bmatrix}
b_{11}c_{11} + b_{12}c_{21} & b_{11}c_{12} + b_{12}c_{22} \\
b_{21}c_{11} + b_{22}c_{21} & b_{21}c_{12} + b_{22}c_{22}
\end{bmatrix}
\]
3. **Compute \( A(BC) \):**
To multiply \( A \) with the resulting matrix \( BC \), perform the following:
\[
A(BC) = \begin{bmatrix}
a_{11}(b_{11}c_{11} + b_{12}c_{21}) + a_{12}(b_{21}c_{11} + b_{22}c_{21}) & a_{11}(b_{11}c_{12} + b_{12}c_{22}) + a_{12}(b_{21}c_{12} + b_{22}c_{22}) \\
a_{21}(b_{11}c_{11} + b_{12}c_{21}) + a_{22}(b_{21}c_{11} + b_{22}c_{21}) & a_{21](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2192a9f0-1ae9-4ed3-a35b-fc11f1172fc0%2Fa860d34d-f5c4-49c3-b43e-6a6eb330fb6f%2F0ag4f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem: Matrix Associativity**
Show by direct calculation for 2×2 matrices \( A, B, C \) that \( A(BC) = (AB)C \).
To solve this problem, we'll need to verify the associative property of matrix multiplication for 2x2 matrices using direct calculation. Associative property in matrices means that the order in which the matrices are grouped does not affect the product.
1. **Define the matrices:**
- Let \( A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \),
- \( B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} \),
- \( C = \begin{bmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{bmatrix} \).
2. **Compute \( BC \):**
\[
BC = \begin{bmatrix}
b_{11}c_{11} + b_{12}c_{21} & b_{11}c_{12} + b_{12}c_{22} \\
b_{21}c_{11} + b_{22}c_{21} & b_{21}c_{12} + b_{22}c_{22}
\end{bmatrix}
\]
3. **Compute \( A(BC) \):**
To multiply \( A \) with the resulting matrix \( BC \), perform the following:
\[
A(BC) = \begin{bmatrix}
a_{11}(b_{11}c_{11} + b_{12}c_{21}) + a_{12}(b_{21}c_{11} + b_{22}c_{21}) & a_{11}(b_{11}c_{12} + b_{12}c_{22}) + a_{12}(b_{21}c_{12} + b_{22}c_{22}) \\
a_{21}(b_{11}c_{11} + b_{12}c_{21}) + a_{22}(b_{21}c_{11} + b_{22}c_{21}) & a_{21
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