2. Let A = and B = al. Show that det(A + B) = det A + det B if and only if a + d = 0.
2. Let A = and B = al. Show that det(A + B) = det A + det B if and only if a + d = 0.
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 42:**
Consider the matrices \( A \) and \( B \) defined as follows:
\[ A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]
\[ B = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]
The task is to prove that:
\[ \det(A + B) = \det(A) + \det(B) \]
if and only if \( a + d = 0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb575d4aa-617f-4320-be65-135468ab142e%2F8a40558e-bc25-4152-96ff-816d994a15d4%2F6sob6xo_processed.png&w=3840&q=75)
Transcribed Image Text:**Question 42:**
Consider the matrices \( A \) and \( B \) defined as follows:
\[ A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]
\[ B = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]
The task is to prove that:
\[ \det(A + B) = \det(A) + \det(B) \]
if and only if \( a + d = 0 \).
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