8. When we explored matrix multiplication in Section 2.2, we saw that some properties that are true for real numbers are not true for matrices. This exercise will investigate that in some more depth. a. Suppose that A and B are two matrices and that AB = 0. If B # 0, what can you say about the linear independence of the columns of A? b. Suppose that we have matrices A, B and C such that AB = AC. We have seen that we cannot generally conclude that B = C. If we assume additionally that A is a matrix whose columns are linearly independent, explain why B = C. You may wish to begin by rewriting the equation AB = AC as AB – AC = A(B – C) = 0.

8. When we explored matrix multiplication in Section 2.2, we saw that some properties that are true for real numbers are not true for matrices. This exercise will investigate that in some more depth. a. Suppose that A and B are two matrices and that AB = 0. If B # 0, what can you say about the linear independence of the columns of A? b. Suppose that we have matrices A, B and C such that AB = AC. We have seen that we cannot generally conclude that B = C. If we assume additionally that A is a matrix whose columns are linearly independent, explain why B = C. You may wish to begin by rewriting the equation AB = AC as AB – AC = A(B – C) = 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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Concept explainers

Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.

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please prove using examples and language of proofs.
8. When we explored matrix multiplication in Section 2.2, we saw that some
properties that are true for real numbers are not true for matrices. This
exercise will investigate that in some more depth.
a. Suppose that A and B are two matrices and that AB = 0. If B+ 0, what
can you say about the linear independence of the columns of A?
b. Suppose that we have matrices A, B and C such that AB = AC. We
have seen that we cannot generally conclude that B = C. If we assume
additionally that A is a matrix whose columns are linearly independent,
explain why B = C. You may wish to begin by rewriting the equation
AB = AC as AB – AC = A(B – C) = 0.

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