Question

Algebra 2 question? Matrices question

Answer 1

Part a
This reference on Matrix multiplication tells us that when we multiply a `1 xx2` matrix by a `2 xx 4` matrix we will obtain a `1xx4` matrix. It is important to know the dimensions of the resulting matrix because you will, then, know how many rows and columns of `c_(ij)s` that you must compute. You start with `c_11, c_12, c_13,..., c_1j` then do the same with row 2, row 3, and so on:

In this case, there is only 1 row with 4 columns:

`[(13, 19)]*[(23,21,19,17),(19,13,11,9)]=[(660 , 520 , 456 , 392)]`

Part b

The meaning of the entries in the resulting matrix is the amount of money spent on T-shirts sold in the 4 months, May, June, July, and August, respectively.

We can see that amount spent in May is `$660` and the amount spent August is `$392`; the difference is:

`$660 - $392 = $268`

Answer 2

`$268` more was spent.

Explanation:

Before you can multiply matrices you have to check whether they are compatible:

A `" " color(red)(1) xx color(blue)(2)" "` matrix and a `" " color(blue)(2) xx color(red)(4)" "` matrix are compatible because the number of columns in the first matrix is the same as the number of rows in the second matrix. This is shown in `color(blue)("blue")`

The answer matrix will be a `" " color(red)(1 xx 4)" "` matrix.

To multiply the matrices:

The values in the row of the first matrix,`[(13,19)]`, must be multiplied by the values in each of the columns of the second matrix.

The products must be added to get the final answer.

`[(13, 19)]*[(color(magenta)(23),color(limegreen)(21),color(darkorange)(19),color(cyan)(17)),(color(magenta)(19),color(limegreen)(13),color(darkorange)(11),color(cyan)(9))]=[color(magenta)(?) " "color(limegreen)(?)" " color(darkorange)(?)" " color(cyan)(?)]`

`[(13, 19)]*[(color(magenta)(23),21,19,17),(color(magenta)(19),13,11,9)]=[color(magenta)(13*23+19*19) " "color(limegreen)(?)" " color(darkorange)(?)" " color(cyan)(?)]`

`[(13, 19)]*[(color(magenta)(23),21,19,17),(color(magenta)(19),13,11,9)]=[color(magenta)(660) " "color(limegreen)(?)" " color(darkorange)(?)" " color(cyan)(?)]`

`[(13, 19)]*[(23,color(limegreen)(21),19,17),(19,color(limegreen)(13),11,9)]=[(660 , color(limegreen)(13*21+19*13) , color(darkorange)(?) color(cyan)(?))]`

`[(13, 19)]*[(23,color(limegreen)(21),19,17),(19,color(limegreen)(13),11,9)]=[(660 , color(limegreen)(520) , color(darkorange)(?)" " color(cyan)(?))]`

`[(13, 19)]*[(23,21,color(darkorange)(19),17),(19,13,color(darkorange)(11),9)]=[(660 , 520 , color(darkorange)(13*19+19*11) color(cyan)(?))]`

`[(13, 19)]*[(23,21,color(darkorange)(19),17),(19,13,color(darkorange)(11),9)]=[(660 , 520 , color(darkorange)(456)" " color(cyan)(?))]`

`[(13,19)]*[(23,21,19,color(cyan)(17)),(19,13,11,color(cyan)(9))]=[(660 , 520 , 456 " " color(cyan)(13*17+19*9))]`

`[(13, 19)]*[(23,21,19,color(cyan)(17)),(19,13,11,color(cyan)(9))]=[(660 , 520 , 456 color(cyan)(392))]`

The final answer is
`[(13, 19)]*[(23,21,19,17),(19,13,11,9)]=[(660 , 520 , 456, 392)]`

The values in the final `1 xx 4` matrix indicate the total amount of money spent on both types of shirts in the months May, June, July and August respectively.

How much more was spent in `color(magenta)("May")` than in `color(cyan)"August"`?

`[(13, 19)]*[(23,21,19,17),(19,13,11,9)]=[color(magenta)(660) , 520 , 456, color(cyan)(392)]`

`$660-$392 = $268`

When multiplying: work across horizontally in the first matrix and vertically in the second matrix.