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Construct a matrix from labelled information or an entry rule

Start a set of 8
1 OF 6
Rule
\(\displaystyle f_{ij}=4i-j\)
Order
\(\displaystyle 2\times 3\)
Entry
\(\displaystyle f_{23}\)

A spreadsheet fills a 2 by 3 block using the displayed rule, where i is the row number and j is the column number, both counted from 1. Find the entry in row 2 and column 3

Year 12, VCE Units 3 & 4, Chapter 10, 4 difficulty levels, No calculatorBuild a worksheet

Construct a matrix from data or a rule

Fix the row and column order before entering values. For a rule, use one-based row and column indices at each address; for a labelled table, copy the number at each matching pair of labels

\[a_{ij}=2i+j+1\]
  1. Fix the row and column order The first subscript identifies a row and the second subscript identifies a column \(\displaystyle a_{ij}\quad\text{is in row }i\text{, column }j\)
  2. Use one-based indices The first row and first column use the indices 1 and 1 \(\displaystyle a_{11}:\quad i=1,\quad j=1\)
  3. Substitute each address For an entry rule, replace i and j with the row and column numbers at the requested address \(\displaystyle a_{23}=2(2)+3+1=8\)
  4. Keep labels as headings For labelled information, the row and column labels set the order and are not numerical matrix entries \(\displaystyle \text{matrix entries}=\text{values at label intersections}\)

Worked example

A 2 × 3 matrix M has entries defined by the displayed rule, where i and j are one-based row and column indices. Construct M

\[m_{ij}=3+i+2j\]
  1. \(\displaystyle i=1,2\text{ for the rows},\quad j=1,2,3\text{ for the columns}\)Use the row number for i and the column number for j, starting both indices at 1
  2. \(\displaystyle m_{11}=3+1+2(1)=6,\quad m_{12}=3+1+2(2)=8,\quad m_{13}=3+1+2(3)=10\)Substitute the first row index and each column index to obtain the first row
  3. \(\displaystyle m_{21}=3+2+2(1)=7,\quad m_{22}=3+2+2(2)=9,\quad m_{23}=3+2+2(3)=11\)Substitute the second row index and each column index to obtain the second row
  4. \(\displaystyle M=\begin{bmatrix}6&8&10\\7&9&11\end{bmatrix}\)Place each value at the row and column address used in its calculation

Common mistakes

Swapping row and column indices
✕\(\displaystyle a_{23}:\quad i=3,\quad j=2\quad\Rightarrow\quad 2(3)+2+1=9\)✓\(\displaystyle a_{23}:\quad i=2,\quad j=3\quad\Rightarrow\quad 2(2)+3+1=8\)
The first subscript is the row number and the second subscript is the column number
Starting an entry rule at zero
✕\(\displaystyle a_{11}=2(0)+0+1=1\)✓\(\displaystyle a_{11}=2(1)+1+1=4\)
Matrix addresses in this method start at 1, so the first entry uses row 1 and column 1
Entering labels as matrix values
✕\(\displaystyle \text{row and column headings are numerical entries of }A\)✓\(\displaystyle \text{row and column headings stay outside }A\text{ as labels}\)
Labels identify the axes of the table, while the matrix contains the numerical values at their intersections

Difficulty levels

introTransfer a compact labelled table into a matrix with the supplied axes
standardConstruct a matrix from a simple one-operation index rule
challengeEvaluate signed or cross-term rules and place selected values in a larger labelled matrix
examChoose the matrix that preserves both axes or satisfies the stated entry rule

Questions about this topic

Why does the first subscript identify the row?+

The two subscripts give an entry's ordered address: the first gives its row and the second gives its column

Why are the first row and column indexed by 1?+

The construction uses one-based matrix addresses, so the top-left entry is in row 1 and column 1

What happens to labels when table data is put into a matrix?+

The labels determine the row and column order but remain headings rather than becoming numerical entries

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, General Mathematics, chapter 10: Matrices

In this chapter

State the order of a matrix and locate its elementsIdentify row, column, square, symmetric and zero matricesRead information from a labelled data matrixRead, transpose and classify a matrixConstruct a matrix from labelled information or an entry rule10A Matrix notation, types and construction: Solve a matrix equation10A Matrix notation, types and construction: Construct, apply and invert a permutation matrixAdd and subtract matricesFind unknown entries in a matrix equationMultiply a matrix by a scalarCombine scalar multiples of matrices10B Linear operations on matrices: Determine whether an inverse exists and calculate itDecide whether a matrix product is defined and give its orderMultiply matricesEvaluate powers of a square matrixIdentify and construct the identity matrix10C Matrix products, powers and permutations: Construct, apply and invert a permutation matrixFind the determinant of a 2 × 2 matrixFind the inverse of a 2 × 2 matrixWrite simultaneous equations as a matrix equationSolve simultaneous equations using the inverse matrix10D Determinants, inverses and matrix equations: Determine whether an inverse exists and calculate it10D Determinants, inverses and matrix equations: Solve a matrix equationUse matrix multiplication to answer an application questionUse row and column matrices of ones to extract totalsRepresent a network in a matrixIdentify the network a connection matrix describesInterpret connection sums and symmetryCount routes using powers of a connection matrixConstruct and analyse relation matrices