Linear Algebra : Matrix-Matrix Product

Study concepts, example questions & explanations for Linear Algebra

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Example Questions

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Example Question #101 : Matrix Matrix Product

Calculate .

Possible Answers:

is not defined.

Correct answer:

Explanation:

, the transpose of , can be found by transposing rows with columns.

, so

Matrices are multiplied by multiplying each row of the first matrix by each column of the second - that is, by adding the products of the entries in corresponding positions. Thus,

This is a square matrix, so it can be raised to a power. To raise a diagonal matrix to a power, simply raise each number in the main diagonal to that power:

.

Example Question #102 : Matrix Matrix Product

and , where all four variables stand for real quantities.

Which must be true of and regardless of the values of the variables?

Possible Answers:

None of the statements given in the other choices are correct.

Correct answer:

None of the statements given in the other choices are correct.

Explanation:

Matrices are multiplied by multiplying each row of the first matrix by each column of the second - that is, by adding the products of the entries in corresponding positions. Thus,

 

 

 

is false in general.

 

, the transpose of , is the result of transposing rows and columns:

. so

is false in general.

 

, so

is false in general.

 

, so

is false in general.

 

Thus, none of the four given statements need be true.

Example Question #103 : Matrix Matrix Product

Let

Find .

Possible Answers:

 is undefined.

Correct answer:

Explanation:

is equal to the two-entry column matrix , so , the transpose, is the row matrix

The product of two matrices is calculated by multiplying rows by columns - adding the corresponding entries in the rows of the first matrix by the columns of the second - so

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