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Use row and column matrices of ones to extract totals

Question 1 of 8

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The data matrix \(Q\) gives the numbers of results recorded by several sports teams, with one row for each team and one column for each result type

Row order: Wallabies, Kookaburras, Eagles, Falcons. Column order: Bonus points, Draws

Q = \begin{bmatrix} 1 & 10 \\ 6 & 6 \\ 11 & 1 \\ 4 & 11 \end{bmatrix}

The following row and column matrices of 1s are available

R = \begin{bmatrix} 1 & 1 & 1 & 1 \end{bmatrix}, \qquad C = \begin{bmatrix} 1 \\ 1 \end{bmatrix}

(a)1 mark

Select the ones matrix and side that produce all column totals of \(Q\)

Multiply on the right by the 2 by 1 column \(\begin{bmatrix} 1 \\ 1 \end{bmatrix}\): \(Q\begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 11 \\ 12 \\ 12 \\ 15 \end{bmatrix}\)

Multiply on the left by the 1 by 4 row \(\begin{bmatrix} 1 & 1 & 1 & 1 \end{bmatrix}\): \(\begin{bmatrix} 1 & 1 & 1 & 1 \end{bmatrix}Q = \begin{bmatrix} 22 & 28 \end{bmatrix}\)

\(\begin{bmatrix} 1 & 1 & 1 & 1 \end{bmatrix}Q = \begin{bmatrix} 22 & 28 \end{bmatrix}\) gives the totals for the rows of \(Q\)

Put the 4 by 1 column \(\begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}\) on the left of \(Q\)

(b)2 marks

Use left multiplication by a row of 1s to find all column totals of \(Q\)

Column totals of \(Q\)