a. Proof: Consider a 1 × 1 matrix, A = [ a ] Then, det ( A ) = a Now, we have − A = [ − a ] Then, we get det ( − A ) = − a = − det ( A ) Similarly consider a 3 × 3 matrix as given below: A = [ a b c 0 d e 0 f

Question

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Answer 1

Expert Solution & Answer

Chapter 2.6, Problem 4P

Explanation of Solution

a.

Proof:

Consider a 1×1 matrix,

A=[a]

Then, det(A)=a

Now, we have

−A=[−a]

Then, we get

det(−A)=−a =−det(A)

Similarly consider a 3×3 matrix as given below:

A=[a b c0 d e0 f

Explanation of Solution

b.

Proof:

Let us consider a 2×2 matrix as given below:

A=[a     bc     d]

Then, we get det(A)=ad−bc

Now, we have

−A=[−a    -b−c    -d]

Then, we get

det(−A)=ad−bc             =det(A)

Similarly consider a 4×4 matrix as,

A=[a     b     c      d0     e      f      g0     h      0    �

Explanation of Solution

Generalize the results of part (a.) and (b.):

From part a.) and part b.), we conclude that

For any n×n matrix,

det(−A)=−det(

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Check out a sample textbook solution

See solution

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If there is a non-singular matrix P such as P-1AP=D, matrix A is called a diagonalizable matrix. A, n x n square matrix is diagonalizable if and only if matrix A has n linearly independent eigenvectors. In this case, the diagonal elements of the diagonal matrix D are the eigenvalues of the matrix A. A=({{1, -1, -1}, {1, 3, 1}, {-3, 1, -1}}) : 1 -1 -1 1 3 1 -3 1 -1 a)Write a program that calculates the eigenvalues and eigenvectors of matrix A using NumPy. b)Write the program that determines whether the D matrix is diagonal by calculating the D matrix, using NumPy. #UsePython

USING PYTHON A tridiagonal matrix is one where the only nonzero elements are the ones on the main diagonal (i.e., ai,j where j = i) and the ones immediately above and belowit(i.e.,ai,j wherej=i+1orj=i−1). Write a function that solves a linear system whose coefficient matrix is tridiag- onal. In this case, Gauss elimination can be made much more efficient because most elements are already zero and don’t need to be modified or added. Please show steps and explain.

Let A be an m × n matrix with m > n. (a) What is the maximum number of nonzero singular values that A can have? (b) If rank(A) = k, how many nonzero singular values does A have?

Answer 2

Expert Solution & Answer

Chapter 2.6, Problem 4P

Explanation of Solution

a.

Proof:

Consider a 1×1 matrix,

A=[a]

Then, det(A)=a

Now, we have

−A=[−a]

Then, we get

det(−A)=−a =−det(A)

Similarly consider a 3×3 matrix as given below:

A=[a b c0 d e0 f

Explanation of Solution

b.

Proof:

Let us consider a 2×2 matrix as given below:

A=[a     bc     d]

Then, we get det(A)=ad−bc

Now, we have

−A=[−a    -b−c    -d]

Then, we get

det(−A)=ad−bc             =det(A)

Similarly consider a 4×4 matrix as,

A=[a     b     c      d0     e      f      g0     h      0    �

Explanation of Solution

Generalize the results of part (a.) and (b.):

From part a.) and part b.), we conclude that

For any n×n matrix,

det(−A)=−det(

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 3

Expert Solution & Answer

Chapter 2.6, Problem 4P

Explanation of Solution

a.

Proof:

Consider a 1×1 matrix,

A=[a]

Then, det(A)=a

Now, we have

−A=[−a]

Then, we get

det(−A)=−a =−det(A)

Similarly consider a 3×3 matrix as given below:

A=[a b c0 d e0 f

Explanation of Solution

b.

Proof:

Let us consider a 2×2 matrix as given below:

A=[a     bc     d]

Then, we get det(A)=ad−bc

Now, we have

−A=[−a    -b−c    -d]

Then, we get

det(−A)=ad−bc             =det(A)

Similarly consider a 4×4 matrix as,

A=[a     b     c      d0     e      f      g0     h      0    �

Explanation of Solution

Generalize the results of part (a.) and (b.):

From part a.) and part b.), we conclude that

For any n×n matrix,

det(−A)=−det(

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 4

Expert Solution & Answer

Chapter 2.6, Problem 4P

Explanation of Solution

a.

Proof:

Consider a 1×1 matrix,

A=[a]

Then, det(A)=a

Now, we have

−A=[−a]

Then, we get

det(−A)=−a =−det(A)

Similarly consider a 3×3 matrix as given below:

A=[a b c0 d e0 f

Explanation of Solution

b.

Proof:

Let us consider a 2×2 matrix as given below:

A=[a     bc     d]

Then, we get det(A)=ad−bc

Now, we have

−A=[−a    -b−c    -d]

Then, we get

det(−A)=ad−bc             =det(A)

Similarly consider a 4×4 matrix as,

A=[a     b     c      d0     e      f      g0     h      0    �

Explanation of Solution

Generalize the results of part (a.) and (b.):

From part a.) and part b.), we conclude that

For any n×n matrix,

det(−A)=−det(

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 5

Expert Solution & Answer

Answer 6

Expert Solution & Answer

Answer 7

Chapter 2.6, Problem 4P

Explanation of Solution

a.

Proof:

Consider a 1×1 matrix,

A=[a]

Then, det(A)=a

Now, we have

−A=[−a]

Then, we get

det(−A)=−a =−det(A)

Similarly consider a 3×3 matrix as given below:

A=[a b c0 d e0 f

Explanation of Solution

b.

Proof:

Let us consider a 2×2 matrix as given below:

A=[a     bc     d]

Then, we get det(A)=ad−bc

Now, we have

−A=[−a    -b−c    -d]

Then, we get

det(−A)=ad−bc             =det(A)

Similarly consider a 4×4 matrix as,

A=[a     b     c      d0     e      f      g0     h      0    �

Explanation of Solution

Generalize the results of part (a.) and (b.):

From part a.) and part b.), we conclude that

For any n×n matrix,

det(−A)=−det(

Answer 8

Chapter 2.6, Problem 4P

Explanation of Solution

a.

Proof:

Consider a 1×1 matrix,

A=[a]

Then, det(A)=a

Now, we have

−A=[−a]

Then, we get

det(−A)=−a =−det(A)

Similarly consider a 3×3 matrix as given below:

A=[a b c0 d e0 f

Answer 9

Chapter 2.6, Problem 4P

Explanation of Solution

a.

Proof:

Consider a 1×1 matrix,

A=[a]

Then, det(A)=a

Now, we have

−A=[−a]

Then, we get

det(−A)=−a =−det(A)

Similarly consider a 3×3 matrix as given below:

A=[a b c0 d e0 f

Answer 10

Explanation of Solution

b.

Proof:

Let us consider a 2×2 matrix as given below:

A=[a     bc     d]

Then, we get det(A)=ad−bc

Now, we have

−A=[−a    -b−c    -d]

Then, we get

det(−A)=ad−bc             =det(A)

Similarly consider a 4×4 matrix as,

A=[a     b     c      d0     e      f      g0     h      0    �

Answer 11

Explanation of Solution

b.

Proof:

Let us consider a 2×2 matrix as given below:

A=[a     bc     d]

Then, we get det(A)=ad−bc

Now, we have

−A=[−a    -b−c    -d]

Then, we get

det(−A)=ad−bc             =det(A)

Similarly consider a 4×4 matrix as,

A=[a     b     c      d0     e      f      g0     h      0    �

Answer 12

Explanation of Solution

Generalize the results of part (a.) and (b.):

From part a.) and part b.), we conclude that

For any n×n matrix,

det(−A)=−det(

Answer 13

Explanation of Solution

Generalize the results of part (a.) and (b.):

From part a.) and part b.), we conclude that

For any n×n matrix,

det(−A)=−det(

Answer 14

Expert Solution & Answer

Answer 15

Expert Solution & Answer

Answer 16

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Check out a sample textbook solution

See solution

Answer 17

Ch. 2.1 - Prob. 1P Ch. 2.1 - Prob. 2P Ch. 2.1 - Prob. 3P Ch. 2.1 - Prob. 4P Ch. 2.1 - Prob. 5P Ch. 2.1 - Prob. 6P Ch. 2.1 - Prob. 7P Ch. 2.2 - Prob. 1P Ch. 2.3 - Prob. 1P Ch. 2.3 - Prob. 2P

a. Proof: Consider a 1 × 1 matrix, A = [ a ] Then, det ( A ) = a Now, we have − A = [ − a ] Then, we get det ( − A ) = − a = − det ( A ) Similarly consider a 3 × 3 matrix as given below: A = [ a b c 0 d e 0 f

Solution Summary: The author illustrates how a 1times 1 matrix, A=left[aright], and mathrmdet(A)=ad-bc

Explanation of Solution

a.

Proof:

Consider a 1×1 matrix,

A=[a]

Then, det(A)=a

Now, we have

−A=[−a]

Then, we get

det(−A)=−a =−det(A)

Similarly consider a 3×3 matrix as given below:

A=[a b c0 d e0 f

Explanation of Solution

Explanation of Solution

Generalize the results of part (a.) and (b.):

From part a.) and part b.), we conclude that

For any n×n matrix,

det(−A)=−det(

Want to see the full answer?

Chapter 2 Solutions

a. Proof: Consider a 1 × 1 matrix, A = [ a ] Then, det ( A ) = a Now, we have − A = [ − a ] Then, we get det ( − A ) = − a = − det ( A ) Similarly consider a 3 × 3 matrix as given below: A = [ a b c 0 d e 0 f

Solution Summary: The author illustrates how a 1times 1 matrix, A=left[aright], and mathrmdet(A)=ad-bc

Explanation of Solution

a. Proof: Consider a 1×1 matrix, A=[a] Then, det(A)=a Now, we have −A=[−a] Then, we get det(−A)=−a =−det(A) Similarly consider a 3×3 matrix as given below: A=[a b c0 d e0 f

Explanation of Solution

b. Proof: Let us consider a 2×2 matrix as given below: A=[a bc d] Then, we get det(A)=ad−bc Now, we have −A=[−a -b−c -d] Then, we get det(−A)=ad−bc =det(A) Similarly consider a 4×4 matrix as, A=[a b c d0 e f g0 h 0 �

Explanation of Solution

Generalize the results of part (a.) and (b.): From part a.) and part b.), we conclude that For any n×n matrix, det(−A)=−det(

Want to see the full answer?

Chapter 2 Solutions

a.

Proof:

Consider a 1×1 matrix,

A=[a]

Then, det(A)=a

Now, we have

−A=[−a]

Then, we get

det(−A)=−a =−det(A)

Similarly consider a 3×3 matrix as given below:

A=[a b c0 d e0 f

b.

Proof:

Let us consider a 2×2 matrix as given below:

A=[a bc d]

Then, we get det(A)=ad−bc

Now, we have

−A=[−a -b−c -d]

Then, we get

det(−A)=ad−bc =det(A)

Similarly consider a 4×4 matrix as,

A=[a b c d0 e f g0 h 0 �

Generalize the results of part (a.) and (b.):

From part a.) and part b.), we conclude that

For any n×n matrix,

det(−A)=−det(