Q62E Consider聽NXN matrices聽A,B,C an... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q62E (page 293)

ConsiderNXN matricesA,B,C and D such that $r a n k (A) = r a n k [\begin{matrix} A & B \\ C & D \end{matrix}] = n$ . Show that
a. $D = C A^{- 1} B$ and
b. The 2X12 matrix $[\begin{matrix} \det (A) & \det (B) \\ \det (C) & \det (D) \end{matrix}]$ is noninvertible. Hint: Consider the product $[\begin{matrix} I_{n} & 0 \\ - C A^{- 1} & I_{n} \end{matrix}] [\begin{matrix} A & B \\ C & D \end{matrix}]$ .

Short Answer

Expert verified

Therefore,

a) It is proved that $D = C A^{- 1} B$ .

b) Yes, $[\begin{matrix} detA & \det B \\ \det C & \det D \end{matrix}]$ it is non-invertible.

Step by step solution

Matrix Definition

Matrix is aset of numbers arranged in rows and columns so as to form a rectangular array.

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 a 鈥� m by n鈥� matrix, written 鈥� mxn.鈥�

Given

Given matrix,

$r a n k (A) = r a n k [\begin{matrix} A & B \\ C & D \end{matrix}] = n$

(a)Step 3: To show D=CA-1B

$To show, [\begin{matrix} I_{n} & 0 \\ - C A^{- 1} & I_{n} \end{matrix}] [\begin{matrix} A & B \\ C & D \end{matrix}] = [\begin{matrix} A & B \\ 0 & - {CA}^{- 1} B + D \end{matrix}] . Since [\begin{matrix} I_{n} & 0 \\ - {CA}^{- 1} & I_{n} \end{matrix}] Invertible, then rank [\begin{matrix} A & B \\ 0 & - {CA}^{- 1} B + D \end{matrix}] = n This means that - {CA}^{- 1} B + D, So, D = {CA}^{- 1} B .$

(b)Step 4: To show [detAdetBdetCdetD] is non-invertible.

From $D = C A^{- 1} B$ ,

We have,

$detD = \det ({CA}^{- 1} B) detD = \det C \det A^{- 1} \det B \det A \det D = \det B \det C \det A \det D - \det B \det C = 0$

So the matrix is $[\begin{matrix} detA & \det B \\ \det C & \det D \end{matrix}]$ is non-invertible.

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Short Answer

Step by step solution

Matrix Definition

Given

(a)Step 3: To show D=CA-1B

(b)Step 4: To show [detAdetBdetCdetD] is non-invertible.

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91影视

ConsiderNXN matricesA,B,C and D such that rank(A)=rank[ABCD]=n. Show thata.D=CA-1Bandb. The 2X12 matrix[detAdetBdetCdetD] is noninvertible. Hint: Consider the product [In0-CA-1In][ABCD].

Short Answer

Step by step solution

Matrix Definition

Given

(a)Step 3: To show D=CA-1B

(b)Step 4: To show [detAdetBdetCdetD] is non-invertible.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Pure Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.