$1$ . Teresa and Robert attend the same school. $\newline$ They keep a record of the awards they have earned and the points gained. $\newline$ The matrices show the numbers of awards and the points gained for each award. $\newline$ (b) Explain what your answer to (a) represents.

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1

. Teresa and Robert attend the same school.

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They keep a record of the awards they have earned and the points gained.

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The matrices show the numbers of awards and the points gained for each award.

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(b) Explain what your answer to (a) represents.

Define Matrices A and B: Teresa's awards matrix is $A$ and Robert's is $B$ . Let's say $A = \left[ \begin{array}{cc} a_{11} & a_{12} \ a_{21} & a_{22} \end{array} \right]$ and $B = \left[ \begin{array}{cc} b_{11} & b_{12} \ b_{21} & b_{22} \end{array} \right]$ . We need to multiply $A$ by $B$ .
Calculate New Matrix C: To multiply the matrices, we do $(a_{11}*b_{11} + a_{12}*b_{21})$ for the first element of the new matrix C. Let's call this $c_{11}$ .
Combine Teresa's and Robert's Awards: Next, we calculate $a_{11} * b_{12} + a_{12} * b_{22}$ for the second element of the first row of $C$ , which is $c_{12}$ .
Combine Teresa's and Robert's Awards: Next, we calculate $(a_{11}*b_{12} + a_{12}*b_{22})$ for the second element of the first row of $C$ , which is $c_{12}$ .For the first element of the second row of $C$ , we do $(a_{21}*b_{11} + a_{22}*b_{21})$ and call it $c_{21}$ .
Combine Teresa's and Robert's Awards: Next, we calculate $(a_{11}*b_{12} + a_{12}*b_{22})$ for the second element of the first row of $C$ , which is $c_{12}$ .For the first element of the second row of $C$ , we do $(a_{21}*b_{11} + a_{22}*b_{21})$ and call it $c_{21}$ .Finally, we calculate $(a_{21}*b_{12} + a_{22}*b_{22})$ for the second element of the second row of $C$ , which is $c_{22}$ .
Combine Teresa's and Robert's Awards: Next, we calculate $(a_{11}*b_{12} + a_{12}*b_{22})$ for the second element of the first row of $C$ , which is $c_{12}$ .For the first element of the second row of $C$ , we do $(a_{21}*b_{11} + a_{22}*b_{21})$ and call it $c_{21}$ .Finally, we calculate $(a_{21}*b_{12} + a_{22}*b_{22})$ for the second element of the second row of $C$ , which is $c_{22}$ .Now we have the new matrix $C = [c_{11} \; c_{12}; c_{21} \; c_{22}]$ . This matrix represents the total points for the combined awards of Teresa and Robert.
Combine Teresa's and Robert's Awards: Next, we calculate $(a_{11}*b_{12} + a_{12}*b_{22})$ for the second element of the first row of $C$ , which is $c_{12}$ .For the first element of the second row of $C$ , we do $(a_{21}*b_{11} + a_{22}*b_{21})$ and call it $c_{21}$ .Finally, we calculate $(a_{21}*b_{12} + a_{22}*b_{22})$ for the second element of the second row of $C$ , which is $c_{22}$ .Now we have the new matrix $C = [c_{11} \; c_{12}; c_{21} \; c_{22}]$ . This matrix represents the total points for the combined awards of Teresa and Robert.The elements of matrix $C$ , $C$ $1$ , $c_{12}$ , $c_{21}$ , and $c_{22}$ , show the points for each category of awards after combining Teresa's and Robert's records.