Linear Algebra : Matrix-Matrix Product

Study concepts, example questions & explanations for Linear Algebra

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Example Questions

Example Question #31 : Matrix Matrix Product

Let , and .

True or false:  is an example of a valid -factorization of .

Possible Answers:

False, because  is not the right kind of matrix.

False, because  is not the right kind of matrix.

False, because  is not a factorization of .

True

Correct answer:

True

Explanation:

An -factorization is a way of expressing a matrix as a product of two matrices  and . For the factorization to be valid:

1)  must be a Lower triangular matrix - all elements above its main diagonal (upper left corner to lower right corner) must be "0".

2)  must be an Upper triangular matrix - all elements below its main diagonal must be "0". 

3) 

The factorization can be seen to satisfy the first two criteria -  is lower triangular in that there are no nonzero elements above its main diagonal, and  is, analogously, upper triangular. It remains to be shown that 

Multiply each row in  by each column in  - add the products of each element in the former by the corresponding element in the latter - as follows:

All three criteria are met, and  gives a valid -factorization of .

Example Question #32 : Matrix Matrix Product

Let , and 

True or false:  is an example of a valid -factorization of .

Possible Answers:

False, because  is not the right kind of matrix.

True

False, because  is not the right kind of matrix.

False, because  is not a factorization of .

Correct answer:

False, because  is not a factorization of .

Explanation:

An -factorization is a way of expressing a matrix as a product of two matrices  and . For the factorization to be valid:

1)  must be a Lower triangular matrix - all elements above its main diagonal (upper left corner to lower right corner) must be "0".

2)  must be an Upper triangular matrix - all elements below its main diagonal must be "0". 

3) 

The factorization can be seen to satisfy the first two criteria -  is lower triangular in that there are no nonzero elements above its main diagonal, and  is, analogously, upper triangular. It remains to be shown that 

Multiply each row in  by each column in  - add the products of each element in the former by the corresponding element in the latter - as follows:

, so  does not give a valid -factorization of .

Example Question #31 : Matrices

True or False: If  is a square matrix, and , then  is either  or .

Possible Answers:

True

False

Correct answer:

False

Explanation:

For example, if , then , but  itself is not  or .

(If  represented a single real number, then the question would be true, but since  is a matrix, the question is not true anymore.)

Example Question #34 : Matrix Matrix Product

What is dimension criteria to multiply two matrices ?

Possible Answers:

 can only be multiplied if the number of rows in  equals the number of rows in 

 can only be multiplied if the number of columns in  equals the number of columns in 

 can only be multiplied if the number of rows in  equals the number of columns in 

 can only be multiplied if the number of columns in  equals the number of rows in 

Correct answer:

 can only be multiplied if the number of columns in  equals the number of rows in 

Explanation:

If  is an  matrix and  is an  matrix, 

 can only be multiplied if , or the number of columns in  equals the number of rows in .  Otherwise there is a mismatch, and the two matrices can not be multiplied.

Example Question #35 : Matrix Matrix Product

If  is a  matrix and  is a  matrix, can the product  be multiplied?  What about ?

Possible Answers:

 can be multiplied 

 cannot be multiplied

 cannot be multiplied 

 can be multiplied 

 can be multiplied 

 can be multiplied 

 cannot be multiplied 

 cannot be multiplied 

Correct answer:

 can be multiplied 

 cannot be multiplied

Explanation:

If  is an  matrix and  is an  matrix,  can only be multiplied if .

Since  is a  matrix and  is a  matrix,  and  can be multiplied.

 has  rows and  has  columns, therefore  cannot be multiplied.

Example Question #31 : Matrices

If  is a  matrix and  is a  matrix, what are the dimensions of the product ?

Possible Answers:

 cannot be multiplied

Correct answer:

Explanation:

If  is an  matrix and  is an  matrix, the dimensions of are .

In this problem, If  is a  matrix and  is a  matrix, so 

the dimensions of  are .

 

Example Question #32 : Matrices

If  is a  matrix and  is a  matrix, what are the dimensions of the product ?

Possible Answers:

 cannot be multiplied

Correct answer:

Explanation:

If  is an  matrix and  is an  matrix, the dimensions of are .

In this problem, If  is a  matrix and  is a  matrix, so 

the dimensions of  are .

 

Example Question #38 : Matrix Matrix Product

Find the product .

Possible Answers:

 

 cannot be multiplied

Correct answer:

Explanation:

 If  is an  matrix and  is an  matrix, 

 can only be multiplied if , or the number of columns in  equals the number of rows in .  Otherwise there is a mismatch, and the two matrices can not be multiplied.   will be an  matrix

Since  is a  matrix and  is a  matrix, then  can be multiplied and will have the dimensions 

To find the product , you must find the dot product of the rows of  and the columns of 

We find  by finding the dot product of the row  of  and column  of .

 

We find  by finding the dot product of the row  of  and column  of .

We use the same method to find  the rest of the matrix values

Example Question #39 : Matrix Matrix Product

Find the product .

Possible Answers:

 cannot be multiplied

Correct answer:

Explanation:

 If  is an  matrix and  is an  matrix, 

 can only be multiplied if , or the number of columns in  equals the number of rows in .  Otherwise there is a mismatch, and the two matrices can not be multiplied.   will be an  matrix

Since  is a  matrix and  is a  matrix, then  can be multiplied and will have the dimensions

To find the product , you must find the dot product of the rows of  and the columns of 

We find  by finding the dot product of the row  of  and column  of .

 

We find  by finding the dot product of the row  of  and column  of .

We use the same method to find  the rest of the matrix values

Example Question #40 : Matrix Matrix Product

Find the product .

Possible Answers:

 cannot be multiplied

Correct answer:

Explanation:

 If  is an  matrix and  is an  matrix, 

 can only be multiplied if , or the number of columns in  equals the number of rows in .  Otherwise there is a mismatch, and the two matrices can not be multiplied.   will be an  matrix

Since  is a  matrix and  is a  matrix, then  can be multiplied and will have the dimensions 

To find the product , you must find the dot product of the rows of  and the columns of 

We find  by finding the dot product of the row  of  and column  of .

 

We find  by finding the dot product of the row  of  and column  of .

We use the same method to find  the rest of the matrix values

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