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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 \(N\) 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: Kookaburras, Wallabies. Column order: Bonus points, Clean sheets, Draws

N = \begin{bmatrix} 1 & 10 & 4 \\ 6 & 5 & 5 \end{bmatrix}

The following row and column matrices of 1s are available

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

(a)1 mark

Select the ones matrix and side that produce all row totals of \(N\)

Put the 1 by 3 row \(\begin{bmatrix} 1 & 1 & 1 \end{bmatrix}\) on the right of \(N\)

Multiply on the right by the 3 by 1 column \(\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\): \(N\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 15 \\ 16 \end{bmatrix}\)

\(N\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 15 \\ 16 \end{bmatrix}\) gives the totals for the columns of \(N\)

Multiply on the left by the 1 by 2 row \(\begin{bmatrix} 1 & 1 \end{bmatrix}\): \(\begin{bmatrix} 1 & 1 \end{bmatrix}N = \begin{bmatrix} 7 & 15 & 9 \end{bmatrix}\)

(b)2 marks

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

Row totals of \(N\)