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Identify row, column, square, symmetric and zero matrices

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Classify common matrix types

Matrix types are identified from the order and the values in their positions. Check the order first, then test the defining property for the type being considered

\[a_{ij}=a_{ji}\text{ for every }i,j\Rightarrow A\text{ is symmetric}\]
  1. 1Check for one row or one column A row matrix has one row, while a column matrix has one column \(\displaystyle A:1\times n\Rightarrow\text{row matrix},\quad B:m\times1\Rightarrow\text{column matrix}\)
  2. 2Check for a square order A square matrix has the same number of rows and columns \(\displaystyle A:n\times n\Rightarrow\text{square matrix}\)
  3. 3Compare mirrored entries A square matrix is symmetric when every entry equals the entry in the mirrored position across the leading diagonal \(\displaystyle a_{ij}=a_{ji}\text{ for every }i,j\Rightarrow A=A^T\)
  4. 4Check every entry for zero A zero matrix has every entry equal to zero, whatever its order \(\displaystyle A=\begin{bmatrix}0&0\\0&0\end{bmatrix}\Rightarrow\text{zero matrix}\)
Year 12, VCE Units 3 & 4, Chapter 10, 4 difficulty levels, No calculatorBuild a worksheet

Classify common matrix types

Matrix types are identified from the order and the values in their positions. Check the order first, then test the defining property for the type being considered

\[a_{ij}=a_{ji}\text{ for every }i,j\Rightarrow A\text{ is symmetric}\]
  1. Check for one row or one column A row matrix has one row, while a column matrix has one column \(\displaystyle A:1\times n\Rightarrow\text{row matrix},\quad B:m\times1\Rightarrow\text{column matrix}\)
  2. Check for a square order A square matrix has the same number of rows and columns \(\displaystyle A:n\times n\Rightarrow\text{square matrix}\)
  3. Compare mirrored entries A square matrix is symmetric when every entry equals the entry in the mirrored position across the leading diagonal \(\displaystyle a_{ij}=a_{ji}\text{ for every }i,j\Rightarrow A=A^T\)
  4. Check every entry for zero A zero matrix has every entry equal to zero, whatever its order \(\displaystyle A=\begin{bmatrix}0&0\\0&0\end{bmatrix}\Rightarrow\text{zero matrix}\)

Worked example

Identify all types that apply to the matrix: row, column, square, symmetric or zero

\[A=\begin{bmatrix}0&2\\2&0\end{bmatrix}\]
  1. \(\displaystyle A:2\times2\)The matrix has two rows and two columns, so it is square
  2. \(\displaystyle a_{12}=2=a_{21},\quad a_{11}=a_{11},\quad a_{22}=a_{22}\)Every entry agrees with its mirrored entry, so the square matrix is symmetric
  3. \(\displaystyle a_{12}=2\ne0\)Not every entry is zero, so the matrix is not a zero matrix
  4. \(\displaystyle A\text{ is square and symmetric, but not a row matrix, column matrix or zero matrix}\)Its two rows and two columns rule out the one-row and one-column types, while the non-zero entries rule out the zero type

Common mistakes

Mistaking a square matrix for a symmetric matrix
✕\(\displaystyle A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\text{ is symmetric because it is square}\)✓\(\displaystyle A=\begin{bmatrix}1&2\\3&4\end{bmatrix},\quad A\ne A^T\text{, so A is not symmetric}\)
A symmetric matrix must have equal values in every pair of mirrored positions, not merely equal row and column counts
Calling a partly zero matrix a zero matrix
✕\(\displaystyle \begin{bmatrix}0&4\\0&0\end{bmatrix}\text{ is a zero matrix}\)✓\(\displaystyle \begin{bmatrix}0&4\\0&0\end{bmatrix}\text{ is not a zero matrix}\)
A zero matrix has every entry equal to zero, so one non-zero entry is enough to rule it out
Confusing the identity matrix with an all-ones matrix
✕\(\displaystyle I_2=\begin{bmatrix}1&1\\1&1\end{bmatrix}\)✓\(\displaystyle I_2=\begin{bmatrix}1&0\\0&1\end{bmatrix}\)
The identity matrix has 1s on the leading diagonal and 0s elsewhere, so it is not a matrix of all 1s or all 0s

Difficulty levels

introIdentify a plainly distinct row, column, square, symmetric or zero matrix
standardName the type of one matrix while distinguishing it from a near miss
challengeClassify matrices in the presence of a 1 × 1 matrix, an identity matrix and a nearly symmetric matrix
examChoose the type that follows from the displayed matrix and its defining property

Questions about this topic

What is the difference between a square and a symmetric matrix?+

A square matrix has equal row and column counts. A symmetric matrix is square and also has equal values in every pair of mirrored positions

Can a zero matrix be rectangular?+

Yes. A zero matrix can have any order as long as every entry is zero

Why is a 1 × 1 matrix special?+

Its single entry makes it a row, column and square matrix at the same time, so more than one type applies

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