Linear Algebra : Operations and Properties

Study concepts, example questions & explanations for Linear Algebra

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Example Questions

Example Question #1 : Symmetric Matrices

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Correct answer:

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Example Question #1 : Symmetric Matrices

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Correct answer:

Explanation:

Example Question #151 : Linear Algebra

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Example Question #1 : Symmetric Matrices

Possible Answers:

Correct answer:

Explanation:

Example Question #153 : Linear Algebra

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Example Question #154 : Linear Algebra

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Example Question #155 : Linear Algebra

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Example Question #11 : Symmetric Matrices

True or false:  is an example of a skew-symmetric matrix.

Possible Answers:

False

True

Correct answer:

False

Explanation:

A square matrix  is defined to be skew-symmetric if its transpose  - the matrix resulting from interchanging its rows and its columns - is equal to its additive inverse; that is, if

.

Interchanging rows and columns, we see that if 

,

then 

.

 can be determined by changing each element in  to its additive inverse:

, since not every element in corresponding positions is equal; in particular, the three elements in the main diagonal differ.  is not a skew-symmetric matrix.

Example Question #11 : Symmetric Matrices

True or false:  is an example of a skew-symmetric matrix.

Possible Answers:

False

True

Correct answer:

True

Explanation:

A square matrix  is defined to be skew-symmetric if its transpose  - the matrix resulting from interchanging its rows and its columns - is equal to its additive inverse; that is, if

.

Interchanging rows and columns, we see that if 

,

then 

.

We see that each element of  is the additive inverse of the corresponding element in , so , and  is skew-symmetric.

Example Question #158 : Linear Algebra

 is a three-by-three nonsingular skew-symmetric matrix 

Then which of the following must be equal to  ?

Possible Answers:

Correct answer:

Explanation:

A square matrix  is defined to be skew-symmetric if its transpose  - the matrix resulting from interchanging its rows and its columns - is equal to its additive inverse; that is, if

.

Therefore, by substitution, 

 must be equal to the opposite of the three-by-three identity matrix, which is .

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