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True/False: If the statement is false, you must justify why it is false. (a) For nx m matrices A and B, det (AB) # det (A)det (B). (b) For an m x n matrix A, rank(A) is the dimension of the null space of A. (c) The non-pivot columns of a matrix A form a basis for the column space of A. (d) An m x m determinant is defined by determinants of (m-1) x (m-1) submatrices.

True/False: If the statement is false, you must justify why it is false. (a) For nx m matrices A and B, det (AB) # det (A)det (B). (b) For an m x n matrix A, rank(A) is the dimension of the null space of A. (c) The non-pivot columns of a matrix A form a basis for the column space of A. (d) An m x m determinant is defined by determinants of (m-1) x (m-1) submatrices.

Advanced Engineering Mathematics
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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Must answer ALL parts, a through d, as they’re all related
**True/False: If the statement is false, you must justify why it is false.** (a) For \( n \times m \) matrices \( A \) and \( B \), \( \det(AB) \neq \det(A)\det(B) \). (b) For an \( m \times n \) matrix \( A \), \(\text{rank}(A)\) is the dimension of the null space of \( A \). (c) The non-pivot columns of a matrix \( A \) form a basis for the column space of \( A \). (d) An \( m \times m \) determinant is defined by determinants of \((m-1) \times (m-1)\) submatrices.
Transcribed Image Text:**True/False: If the statement is false, you must justify why it is false.** (a) For \( n \times m \) matrices \( A \) and \( B \), \( \det(AB) \neq \det(A)\det(B) \). (b) For an \( m \times n \) matrix \( A \), \(\text{rank}(A)\) is the dimension of the null space of \( A \). (c) The non-pivot columns of a matrix \( A \) form a basis for the column space of \( A \). (d) An \( m \times m \) determinant is defined by determinants of \((m-1) \times (m-1)\) submatrices.
Expert Solution
Step 1

a)

 False-

 

Determinent is defined only for square matrices. 

So for non square matrices the concept of Determinent is not valid. 

 

So whenever n=m the result is true , unless it is not true for all  genaral matrices. 

 

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