[5] [Elementary row operations as matrices] Consider the two matrices A = 2 2 -1 4 7 2 3 1 Ã 2 0 5 0 -5 -23/2/ 3 5/2 (a) Write down two elementary row operations to transform A into Ã. (b) For each elementary row operation, construct a 3 × 3 matrix P such that multiplication of A on the left by P is identical to performing the ele- mentary row operation. Are the two matrices P₁ and P₁ lower triangular? (c) Multiply the two matrices P₁ and P₁ from part (b) together to pro- duce one matrix P₁. Is P₁ also lower triangular?
[5] [Elementary row operations as matrices] Consider the two matrices A = 2 2 -1 4 7 2 3 1 Ã 2 0 5 0 -5 -23/2/ 3 5/2 (a) Write down two elementary row operations to transform A into Ã. (b) For each elementary row operation, construct a 3 × 3 matrix P such that multiplication of A on the left by P is identical to performing the ele- mentary row operation. Are the two matrices P₁ and P₁ lower triangular? (c) Multiply the two matrices P₁ and P₁ from part (b) together to pro- duce one matrix P₁. Is P₁ also lower triangular?
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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Question
Row operations
![**[5] [Elementary row operations as matrices]**
Consider the two matrices
\[
A = \begin{pmatrix} 2 & 2 & 3 \\ -1 & 4 & 1 \\ 7 & 2 & -1 \end{pmatrix}
\quad
\tilde{A} = \begin{pmatrix} 2 & 2 & 3 \\ 0 & 5 & \frac{5}{2} \\ 0 & -5 & -\frac{23}{2} \end{pmatrix}
\]
(a) Write down two elementary row operations to transform \(A\) into \(\tilde{A}\).
(b) For each elementary row operation, construct a \(3 \times 3\) matrix \(P\) such that multiplication of \(A\) on the left by \(P\) is identical to performing the elementary row operation. Are the two matrices \(P_I\) and \(P_{II}\) lower triangular?
(c) Multiply the two matrices \(P_I\) and \(P_{II}\) from part (b) together to produce one matrix \(P_1\). Is \(P_1\) also lower triangular?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2F5c4eb994-b9ec-4fe9-9355-f0d793bf2e4b%2Fbet3g57_processed.png&w=3840&q=75)
Transcribed Image Text:**[5] [Elementary row operations as matrices]**
Consider the two matrices
\[
A = \begin{pmatrix} 2 & 2 & 3 \\ -1 & 4 & 1 \\ 7 & 2 & -1 \end{pmatrix}
\quad
\tilde{A} = \begin{pmatrix} 2 & 2 & 3 \\ 0 & 5 & \frac{5}{2} \\ 0 & -5 & -\frac{23}{2} \end{pmatrix}
\]
(a) Write down two elementary row operations to transform \(A\) into \(\tilde{A}\).
(b) For each elementary row operation, construct a \(3 \times 3\) matrix \(P\) such that multiplication of \(A\) on the left by \(P\) is identical to performing the elementary row operation. Are the two matrices \(P_I\) and \(P_{II}\) lower triangular?
(c) Multiply the two matrices \(P_I\) and \(P_{II}\) from part (b) together to produce one matrix \(P_1\). Is \(P_1\) also lower triangular?
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