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State the order of a matrix and locate its elements

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Matrix A of weekly hours: rows Junior, Senior, Masters; columns Pool, Gym, Track, Rest. Rows are 5,2,3,1; 6,3,2,1; 4,1,2,2A523163214122

A swimming club records weekly hours for three squads across four training types. Rows of A are Junior, Senior then Masters; columns are Pool, Gym, Track then Rest. What is the order of A?

Year 11, VCE Units 1 & 2, Chapter 4, 4 difficulty levels, No calculatorBuild a worksheet

State order and locate elements

The order gives rows first and columns second. In a named element, the first subscript selects the row and the second subscript selects the column

\[\text{order}=m \times n,\quad a_{ij}=\text{entry in row }i\text{, column }j\]
  1. Write rows before columns Count the horizontal rows first and the vertical columns second \(\displaystyle \text{order}=3 \times 2\)
  2. Read the first subscript as the row For an entry, the first subscript i gives the row and the second subscript j gives the column \(\displaystyle a_{23}=\text{row 2, column 3}\)
  3. Count entries by multiplication Multiply the number of rows by the number of columns to find the number of entries \(\displaystyle 3 \times 2=6\)
  4. Do not count labels as entries Only the data inside the matrix is included in the order and entry count \(\displaystyle \text{data order}=2 \times 3\)

Worked example

The matrix B is given below. State its order, count its entries and find the requested element

\[B=\begin{bmatrix}5&8&1\\4&7&2\end{bmatrix},\quad\text{find }b_{23}\]
  1. \(\displaystyle \text{order of }B=2 \times 3\)Matrix B has two rows and three columns
  2. \(\displaystyle 2 \times 3=6\)The matrix therefore has 6 entries
  3. \(\displaystyle b_{23}=\text{row 2, column 3}\)The first subscript selects row 2 and the second selects column 3
  4. \(\displaystyle b_{23}=2\)The entry in row 2, column 3 is 2

Common mistakes

Writing columns before rows
✕\(\displaystyle \text{order of }S=5 \times 3\)✓\(\displaystyle \text{order of }S=3 \times 5\)
The order is written rows first and columns second, so reversing the dimensions gives the wrong order
Transposing the element subscripts
✕\(\displaystyle s_{32}=120\)✓\(\displaystyle s_{32}=133\)
In the ticket sales matrix S, row 3 and column 2 contains 133. Reversing the subscripts selects row 2 and column 3, which contains 120
Counting labels as entries
✕\(\displaystyle \text{order of }S=4 \times 6\)✓\(\displaystyle \text{order of }S=3 \times 5\)
The ticket sales matrix S has 3 data rows and 5 data columns. Its row and column labels identify the data but are not entries in the matrix

Difficulty levels

introUse a small non-square matrix to state its order and find one element or just its order
standardRead labelled rows and columns in both address directions and state requested elements
challengeFind repeated-value elements and count all entries as well as stating the order
examChoose the correct order or element value from alternatives containing subscript and dimension errors

Questions about this topic

Why are rows written first in the order?+

The convention for matrix order is rows multiplied by columns, so a matrix with m rows and n columns has order m by n

How do the two subscripts identify an entry?+

It means the entry in row i and column j. The row number is read before the column number

How is the number of entries found?+

Multiply the number of rows by the number of columns. A 2 by 3 matrix has 6 entries

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 4: Matrices

In this chapter

State the order of a matrix and locate its elements4A Matrix types and order: Identify row, column, square, symmetric and zero matricesRead information from a labelled data matrix4B Matrix addition and subtraction: Identify row, column, square, symmetric and zero matricesAdd and subtract matrices4B Matrix addition and subtraction: Find unknown entries in a matrix equationMultiply a matrix by a scalarCombine scalar multiples of matricesDecide whether a matrix product is defined and give its orderMultiply matrices4D Matrix products and powers: Evaluate powers of a square matrixIdentify and construct the identity matrix4E Inverse matrices and simultaneous equations: Find unknown entries in a matrix equationFind 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 matrix4F Matrices for networks: Evaluate powers of a square matrixRepresent a network in a matrixIdentify the network a connection matrix describesInterpret connection sums and symmetryCount routes using powers of a connection matrixSet up a transition matrixUse a transition matrix to predict the next step4H State matrices and recurrence: Evaluate powers of a square matrixGenerate state matrices by recursionDetermine the long-run state of a transition modelUse matrix multiplication to answer an application questionUse row and column matrices of ones to extract totals