Reading and transposing matrices
Matrix order and element addresses use rows first and columns second. A transpose swaps those two addresses, and a matrix type is checked from its defining condition
\[A_{m \times n}\longrightarrow A^{T}_{n \times m},\qquad a_{ij}\longrightarrow a^{T}_{ji}\]
- State order as rows by columns Count horizontal rows first and vertical columns second \(\displaystyle A=\begin{bmatrix}2&-1&3\\0&4&5\end{bmatrix}\Rightarrow\text{ order }2 \times 3\)
- Read an element address For an entry, the first subscript gives the row and the second gives the column \(\displaystyle a_{23}\text{ is in row }2\text{, column }3\)
- Swap addresses for a transpose The entry at row i and column j moves to row j and column i in the transpose \(\displaystyle A_{2 \times 3}\Rightarrow A^{T}_{3 \times 2}\)
- Test the defining property A special matrix is identified from its order, zero entries or equal mirrored entries, as required by its definition \(\displaystyle d_{12}=0,\quad d_{21}=0\text{ for a diagonal matrix}\)
Worked example
For the matrix A below, state the order of A, form A transpose and find its entry in row 3, column 2
\[A=\begin{bmatrix}2&-1&3\\0&4&5\end{bmatrix}\]
- \(\displaystyle A\text{ has order }2 \times 3\)There are 2 horizontal rows and 3 vertical columns
- \(\displaystyle A^{T}\text{ has order }3 \times 2\)Transposing changes the rows by columns order from 2 by 3 to 3 by 2.
- \(\displaystyle (A^{T})_{32}=A_{23}\)The transpose address 3,2 is taken from the original address 2,3.
- \(\displaystyle (A^{T})_{32}=5\)The value in row 2, column 3 of A is 5
Common mistakes
Copying the matrix instead of transposing
✕\(\displaystyle A^{T}=\begin{bmatrix}2&-1&3\\0&4&5\end{bmatrix}\)✓\(\displaystyle A^{T}=\begin{bmatrix}2&0\\-1&4\\3&5\end{bmatrix}\)
A transpose swaps every row and column address, so the order changes from 2 by 3 to 3 by 2Keeping the original order
✕\(\displaystyle A_{2 \times 3}\Rightarrow A^{T}_{2 \times 3}\)✓\(\displaystyle A_{2 \times 3}\Rightarrow A^{T}_{3 \times 2}\)
The original rows become columns and the original columns become rowsChecking only one side of a diagonal
✕\(\displaystyle D=\begin{bmatrix}1&2\\0&3\end{bmatrix}\text{ is diagonal because }d_{21}=0\)✓\(\displaystyle D=\begin{bmatrix}1&2\\0&3\end{bmatrix}\text{ is not diagonal because }d_{12}=2\ne0\)
A diagonal matrix has zero in every off-diagonal position, not only below or above the leading diagonalDifficulty levels
introState matrix order and read a named element from a small non-square matrix
standardTranspose a matrix and test a named structural property
challengeHandle overlapping matrix properties and a near-miss structural entry
examChoose a transpose from copied, wrong-order and wrong-diagonal matrices
Questions about this topic
How is matrix order written?+
Matrix order is rows by columns. A matrix with 2 rows and 3 columns has order 2 by 3
Which entry does a subscript pair identify?+
The first subscript i gives the row and the second subscript j gives the column. The pair identifies a position in the matrix
How is a transpose formed?+
Interchange the row and column of every entry. A matrix of order m by n becomes one of order n by m
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