Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q43E (page 308)

Show that $A (adjA) = 0 = (adjA) A$ for all noninvertible $n 脳 n$ matrices A. See Exercise 42.

Short Answer

Expert verified

Therefore,

$A (adjA) = (adjA) A = 0$ and it is showed.

Step by step solution

Matrix Definition.

Matrix is a set of numbersarranged in rows and columnsso as to form a rectangulararray.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an 鈥渕 by n鈥� matrix, written 鈥�. $m 脳 n$ 鈥�

To show that A(adjA)=0=(adjA)A.

Let,

$B = a d j A .$

Then, the entry in the i -th row and j -th column of $(A (a d j A))$ equals

${鈭�}_{j = 1}^{n} a_{i, j}, b j i = {鈭�}_{j = 1}^{n} a_{i, j} {(- 1)}^{i + j} d e t A_{i, j}$

which is, in fact, the Laplace expansion of the determinant along the i -th row, and as such, equals 0 .

Therefore,

$A (a d j A) = (a d j A) A = 0 .$

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91影视

Show that $A (adjA) = 0 = (adjA) A$ for all noninvertible $n 脳 n$ matrices A. See Exercise 42.

Short Answer

Step by step solution

Matrix Definition.

To show that A(adjA)=0=(adjA)A.

One App. One Place for Learning.

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91影视

Show that A(adjA)=0=(adjA)Afor all noninvertible n脳nmatrices A. See Exercise 42.

Short Answer

Step by step solution

Matrix Definition.

To show that A(adjA)=0=(adjA)A.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Applied Mathematics

Theoretical and Mathematical Physics

Pure Maths

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Mechanics Maths

Study anywhere. Anytime. Across all devices.

Show that $A (adjA) = 0 = (adjA) A$ for all noninvertible $n 脳 n$ matrices A. See Exercise 42.