# Linear Algebra : Matrix-Matrix Product

## Example Questions

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### Example Question #101 : Matrix Matrix Product

Calculate .

is not defined.

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?

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

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 .

is undefined.

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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