Correct statements A, B, C, and D to make them accurate. **A.** A $5 \times 7$ matrix may have up to seven linearly independent columns.**B.** If $A$ is a $65 \times 17$ matrix of rank 10, then there are 10 linearly independent vectors that satisfy $A^T y = 0$.**C.** Let $A$ be a $65 \times 17$ matrix of rank 17. Then if $A^T y = b$ is consistent, it has a unique solution.**D.** Let $A$ be a $65 \times 17$ matrix of rank 17. Then if $Ax = b$ is consistent, it has infinitely many solutions.

(A) Given a matrix A of order 5×7 Given statement: A 5×7 matrix may have up to seven linearly…

Answered: A. A 5 x 7 matrix may have up to seven…

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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Correct statements A, B, C, and D to make them accurate.

**A.** A $5 \times 7$ matrix may have up to seven linearly independent columns.

**B.** If $A$ is a $65 \times 17$ matrix of rank 10, then there are 10 linearly independent vectors that satisfy $A^T y = 0$.

**C.** Let $A$ be a $65 \times 17$ matrix of rank 17. Then if $A^T y = b$ is consistent, it has a unique solution.

**D.** Let $A$ be a $65 \times 17$ matrix of rank 17. Then if $Ax = b$ is consistent, it has infinitely many solutions.

Expert Solution

Step 1

(A)

Given a matrix $A$ of order $5 \times 7$

Given statement: $A$ $5 \times 7$ matrix may have up to seven linearly independent columns

This statement is not correct,

Suppose if $A$ has 7 linearly independent columns, then the determinant of the matrix $A$ is non zero, but since the order $5 \times 7$ is not symmetric so determinant of this matrix is not defined

Therefore the number of linearly independent columns is not 7.

Similar is the case for 6 linearly independent columns.

But matrix $A$ of order $5 \times 7$ contains a minor of order 5,

If the minor of order 5 is non zero then the matrix of an order $5 \times 7$ may have 5 linearly independent columns

Therefore a matrix $A$ of order $5 \times 7$ may have up to 5 linearly independent columns.

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