Compute AB by block multiplication, using the indicated partitioning. AB= -1 00 -- A = 0 1 00 B = 0 0 2 3 000 1 000 23 0 -1 1 0 00 1 00 1
Compute AB by block multiplication, using the indicated partitioning. AB= -1 00 -- A = 0 1 00 B = 0 0 2 3 000 1 000 23 0 -1 1 0 00 1 00 1
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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![**Title: Block Matrix Multiplication Explained**
**Objective:**
Learn how to compute the product of two matrices using block multiplication and the indicated partitioning.
**Matrix Multiplication by Block Partitioning Example:**
Consider matrices \( A \) and \( B \) given as:
\[
A =
\begin{bmatrix}
1 & -1 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 3
\end{bmatrix}
,\quad
B =
\begin{bmatrix}
2 & 3 & 0 \\
-1 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 1
\end{bmatrix}
\]
**Instruction:**
Compute \( AB \) by block multiplication, using the indicated partitioning.
**Block Partitioning Layout:**
The process involves creating smaller submatrices (blocks) to simplify the calculations. In the diagram:
\[
AB =
\begin{bmatrix}
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{}
\end{bmatrix}
\]
It indicates that matrices \( A \) and \( B \) are partitioned into blocks to handle calculations in smaller parts. Here's how you can proceed:
1. **Partition Matrix A:**
- The first two rows and first two columns form one \( 2 \times 2 \) block.
- The remaining values form the second \( 1 \times 2 \) and third \( 1 \times 2 \) blocks.
2. **Partition Matrix B:**
- The first two rows and first two columns form one \( 2 \times 2 \) block.
- The remaining values form additional blocks corresponding to their parts in matrix B.
**Step-by-Step Block Multiplication:**
- Denote the partitioned blocks of \( A \) and \( B \) as:
- \( A_{ij} \)
- \( B_{ij} \)
- Calculate the product using the blocks:
\[
AB_{ij} = \text{sum of products of corresponding blocks of } A \text{ and } B
\]
**Example Calculation:**
\[
\begin{bmatrix](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F715e5f42-2b41-4404-933d-923c40f8873c%2F9ee45b96-5357-4f1a-a40a-863d494a0588%2Fd2zu72s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Block Matrix Multiplication Explained**
**Objective:**
Learn how to compute the product of two matrices using block multiplication and the indicated partitioning.
**Matrix Multiplication by Block Partitioning Example:**
Consider matrices \( A \) and \( B \) given as:
\[
A =
\begin{bmatrix}
1 & -1 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 3
\end{bmatrix}
,\quad
B =
\begin{bmatrix}
2 & 3 & 0 \\
-1 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 1
\end{bmatrix}
\]
**Instruction:**
Compute \( AB \) by block multiplication, using the indicated partitioning.
**Block Partitioning Layout:**
The process involves creating smaller submatrices (blocks) to simplify the calculations. In the diagram:
\[
AB =
\begin{bmatrix}
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{}
\end{bmatrix}
\]
It indicates that matrices \( A \) and \( B \) are partitioned into blocks to handle calculations in smaller parts. Here's how you can proceed:
1. **Partition Matrix A:**
- The first two rows and first two columns form one \( 2 \times 2 \) block.
- The remaining values form the second \( 1 \times 2 \) and third \( 1 \times 2 \) blocks.
2. **Partition Matrix B:**
- The first two rows and first two columns form one \( 2 \times 2 \) block.
- The remaining values form additional blocks corresponding to their parts in matrix B.
**Step-by-Step Block Multiplication:**
- Denote the partitioned blocks of \( A \) and \( B \) as:
- \( A_{ij} \)
- \( B_{ij} \)
- Calculate the product using the blocks:
\[
AB_{ij} = \text{sum of products of corresponding blocks of } A \text{ and } B
\]
**Example Calculation:**
\[
\begin{bmatrix
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