1. Suppose A is a 3x3 matrix. Suppose det A = 12 which are necessarily true ? a) AX-B has an infinite # of soins b) A is non-singular c) A is row equivalent to d) A has a column of O's. e) det (IAI)=12 f) det (A-¹A) = 1 S O 050 005
1. Suppose A is a 3x3 matrix. Suppose det A = 12 which are necessarily true ? a) AX-B has an infinite # of soins b) A is non-singular c) A is row equivalent to d) A has a column of O's. e) det (IAI)=12 f) det (A-¹A) = 1 S O 050 005
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 1: Analysis of a 3x3 Matrix**
**Given:**
- Suppose \( A \) is a \( 3 \times 3 \) matrix.
- Suppose \(\det A = 12\).
**Question:**
Which of the following statements are necessarily true?
a) The equation \( AX = B \) has an infinite number of solutions.
b) \( A \) is non-singular.
c) \( A \) is row equivalent to
\[
\begin{bmatrix}
5 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 5
\end{bmatrix}
\]
d) \( A \) has a column of zeros.
e) \(\det (IAI) = 12\).
f) \(\det (A^{-1}A) = 1\).
**Explanation:**
- Statement a) relates to the solutions of the system of equations.
- Statement b) refers to the non-singularity property of the matrix.
- Statement c) considers row equivalence with a diagonal matrix.
- Statement d) deals with the presence of a zero column.
- Statement e) involves an identity operation on the determinant.
- Statement f) addresses the determinant of a product of \( A \) and its inverse.
The matrix provided in c) is a 3x3 diagonal matrix with all diagonal elements equal to 5.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39ee8330-8a26-4cdb-b3ef-e7a0164aa18c%2F73df1442-c33a-4697-adbc-6ee41b274e02%2Fcs85jye_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 1: Analysis of a 3x3 Matrix**
**Given:**
- Suppose \( A \) is a \( 3 \times 3 \) matrix.
- Suppose \(\det A = 12\).
**Question:**
Which of the following statements are necessarily true?
a) The equation \( AX = B \) has an infinite number of solutions.
b) \( A \) is non-singular.
c) \( A \) is row equivalent to
\[
\begin{bmatrix}
5 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 5
\end{bmatrix}
\]
d) \( A \) has a column of zeros.
e) \(\det (IAI) = 12\).
f) \(\det (A^{-1}A) = 1\).
**Explanation:**
- Statement a) relates to the solutions of the system of equations.
- Statement b) refers to the non-singularity property of the matrix.
- Statement c) considers row equivalence with a diagonal matrix.
- Statement d) deals with the presence of a zero column.
- Statement e) involves an identity operation on the determinant.
- Statement f) addresses the determinant of a product of \( A \) and its inverse.
The matrix provided in c) is a 3x3 diagonal matrix with all diagonal elements equal to 5.
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