Linear Algebra : Operations and Properties

Study concepts, example questions & explanations for Linear Algebra

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Example Questions

Example Question #7 :Symmetric Matrices

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Example Question #8 :Symmetric Matrices

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Example Question #71 :Operations And Properties

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Example Question #10 :Symmetric Matrices

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Example Question #11 :Symmetric Matrices

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Example Question #11 :Symmetric Matrices

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Example Question #151 :Linear Algebra

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Example Question #14 :Symmetric Matrices

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

Possible Answers:

False

True

Correct answer:

False

Explanation:

A square matrixis 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 into 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 #15 :Symmetric Matrices

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

Possible Answers:

False

True

Correct answer:

True

Explanation:

A square matrixis 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 ofis the additive inverse of the corresponding element in, so, andis skew-symmetric.

Example Question #16 :Symmetric Matrices

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

Then which of the followingmustbe equal to?

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Correct answer:

Explanation:

A square matrixis 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,

必须等于相反of the three-by-three identity matrix, which is.

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