Assume that the matrices below are partitioned conformably for block multiplication. Compute the product. O I P X I O Cs
Assume that the matrices below are partitioned conformably for block multiplication. Compute the product. O I P X I O Cs
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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![### Block Matrix Multiplication Example
**Problem:**
Assume that the matrices below are partitioned conformably for block multiplication. Compute the product.
\[
\left[ \begin{array}{cc}
0 & I \\
I & 0
\end{array} \right]
\left[ \begin{array}{cc}
P & X \\
C & S
\end{array} \right]
\]
**Solution:**
To multiply these block matrices, follow the rules of block matrix multiplication. Each element of the resulting matrix is derived by multiplying the corresponding blocks and summing the resulting blocks:
Given matrices:
\[ A = \left[ \begin{array}{cc}
0 & I \\
I & 0
\end{array} \right], \quad B = \left[ \begin{array}{cc}
P & X \\
C & S
\end{array} \right] \]
The resulting matrix \( AB \) can be found by the formula:
\[ AB = \left[ \begin{array}{cc}
A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \\
A_{21}B_{11} + A_{22}B_{21} & A_{21}B_{12} + A_{22}B_{22}
\end{array} \right] \]
For the given matrices:
\[ A_{11} = 0, \quad A_{12} = I, \quad A_{21} = I, \quad A_{22} = 0 \]
Substituting these values into the formula, we get:
\[
\left[ \begin{array}{cc}
0 \cdot P + I \cdot C & 0 \cdot X + I \cdot S \\
I \cdot P + 0 \cdot C & I \cdot X + 0 \cdot S
\end{array} \right]
=
\left[ \begin{array}{cc}
C & S \\
P & X
\end{array} \right]
\]
So, the product of the block matrices is:
\[
\left[ \begin{array}{cc}
C & S \\
P & X
\end{array](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F13b3ce4d-7939-40c7-a8c2-01af91160cea%2F2833b52d-e4df-4c9f-8761-fdd1f62771ad%2F4wg2bxv_processed.png&w=3840&q=75)
Transcribed Image Text:### Block Matrix Multiplication Example
**Problem:**
Assume that the matrices below are partitioned conformably for block multiplication. Compute the product.
\[
\left[ \begin{array}{cc}
0 & I \\
I & 0
\end{array} \right]
\left[ \begin{array}{cc}
P & X \\
C & S
\end{array} \right]
\]
**Solution:**
To multiply these block matrices, follow the rules of block matrix multiplication. Each element of the resulting matrix is derived by multiplying the corresponding blocks and summing the resulting blocks:
Given matrices:
\[ A = \left[ \begin{array}{cc}
0 & I \\
I & 0
\end{array} \right], \quad B = \left[ \begin{array}{cc}
P & X \\
C & S
\end{array} \right] \]
The resulting matrix \( AB \) can be found by the formula:
\[ AB = \left[ \begin{array}{cc}
A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \\
A_{21}B_{11} + A_{22}B_{21} & A_{21}B_{12} + A_{22}B_{22}
\end{array} \right] \]
For the given matrices:
\[ A_{11} = 0, \quad A_{12} = I, \quad A_{21} = I, \quad A_{22} = 0 \]
Substituting these values into the formula, we get:
\[
\left[ \begin{array}{cc}
0 \cdot P + I \cdot C & 0 \cdot X + I \cdot S \\
I \cdot P + 0 \cdot C & I \cdot X + 0 \cdot S
\end{array} \right]
=
\left[ \begin{array}{cc}
C & S \\
P & X
\end{array} \right]
\]
So, the product of the block matrices is:
\[
\left[ \begin{array}{cc}
C & S \\
P & X
\end{array
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