Interpret connection sums and symmetry practice questions | General Mathematics (Year 12, VCE Units 3 & 4, Matrices, Communication and dominance matrices)

Read connection sums and symmetry

A connection matrix records direct links between an ordered list of vertices. Adding a row or column gives a connection count, and an undirected network produces a symmetric matrix

m_{ij}=m_{ji}\text{ for every }i,j\quad\Rightarrow\quad M=M^T

Keep one vertex order Use the same vertex order for the rows and columns of the connection matrix \(\displaystyle m_{ij}=\text{number of direct links from }i\text{ to }j\)
Add along the required axis A row sum counts connections leaving a vertex, while a column sum counts connections arriving at a vertex \(\displaystyle \text{row }i:\sum_jm_{ij},\quad\text{column }j:\sum_im_{ij}\)
Count a loop on the diagonal A loop at a vertex is recorded in its leading-diagonal entry and contributes once to the relevant sum \(\displaystyle m_{ii}=1\quad\Rightarrow\quad\text{one loop at vertex }i\)
Compare mirrored entries If every connection is two-way, every pair of mirrored entries is equal and the matrix is symmetric. One matching pair alone does not establish symmetry \(\displaystyle m_{ij}=m_{ji}\text{ for every }i,j\quad\Rightarrow\quad M=M^T\)

Worked example

The connection matrix for vertices A, B and C is shown below. Find the direct connection count for B and decide whether the matrix is symmetric

M=\begin{bmatrix}0&1&2\\1&0&1\\2&1&0\end{bmatrix}

\(\displaystyle \text{column B}=1+0+1=2\)Adding the B column counts the direct connections arriving at B
\(\displaystyle \text{row B}=1+0+1=2\)Adding the B row also gives 2 because every connection in this matrix is two-way
\(\displaystyle m_{12}=m_{21}=1,\quad m_{13}=m_{31}=2,\quad m_{23}=m_{32}=1\)Each off-diagonal entry agrees with its mirrored entry
\(\displaystyle M=M^T\quad\Rightarrow\quad M\text{ is symmetric}\)The matrix is symmetric, and vertex B has 2 direct connections

Common mistakes

Checking only the diagonal

✕\(\displaystyle m_{11}=m_{22}=m_{33}=0\quad\Rightarrow\quad M=M^T\)✓\(\displaystyle m_{ij}=m_{ji}\text{ for every }i,j\quad\Rightarrow\quad M=M^T\)

Diagonal entries compare with themselves, so symmetry must be checked with every pair of mirrored positions

Treating a connection sum as a vertex count

✕\(\displaystyle \text{column sum}=3\quad\Rightarrow\quad\text{there are 3 vertices}\)✓\(\displaystyle \text{column sum}=3\quad\Rightarrow\quad\text{the vertex has 3 direct connections}\)

A row or column sum adds connection entries for one vertex; it does not count the vertices in the whole network

Mirroring a one-way connection

✕\(\displaystyle M=\begin{bmatrix}0&1\\1&0\end{bmatrix}\quad\text{for the one-way link }A\to B\)✓\(\displaystyle M=\begin{bmatrix}0&1\\0&0\end{bmatrix}\quad\text{for the one-way link }A\to B\)

A one-way link is recorded in one directed position, so it is not copied into the mirrored position

Difficulty levels

introAdd a row or column of an undirected connection matrix and state its meaning

standardUse sums, symmetry and a comparison of connection counts in a larger undirected matrix

challengeDistinguish row and column sums for a directed network and count a loop consistently

examChoose a correct connection sum, meaning or symmetry statement

Questions about this topic

What does a row sum mean in a connection matrix?+

With rows recording the starting vertex, a row sum counts the direct connections leaving that vertex

When is a connection matrix symmetric?+

It is symmetric exactly when every mirrored pair has equal counts: the entry in row i and column j equals the entry in row j and column i for every pair of vertices. Equal row and column sums alone are not sufficient

What does a diagonal entry represent?+

A diagonal entry records a connection from a vertex to itself, such as a loop, and contributes to that vertex's sum

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 elements Identify row, column, square, symmetric and zero matrices Read information from a labelled data matrix Read, transpose and classify a matrix Construct a matrix from labelled information or an entry rule 10A Matrix notation, types and construction: Solve a matrix equation 10A Matrix notation, types and construction: Construct, apply and invert a permutation matrix Add and subtract matrices Find unknown entries in a matrix equation Multiply a matrix by a scalar Combine scalar multiples of matrices 10B Linear operations on matrices: Determine whether an inverse exists and calculate it Decide whether a matrix product is defined and give its order Multiply matrices Evaluate powers of a square matrix Identify and construct the identity matrix 10C Matrix products, powers and permutations: Construct, apply and invert a permutation matrix Find the determinant of a 2 × 2 matrix Find the inverse of a 2 × 2 matrix Write simultaneous equations as a matrix equation Solve simultaneous equations using the inverse matrix 10D Determinants, inverses and matrix equations: Determine whether an inverse exists and calculate it 10D Determinants, inverses and matrix equations: Solve a matrix equation Use matrix multiplication to answer an application question Use row and column matrices of ones to extract totals Represent a network in a matrix Identify the network a connection matrix describes Interpret connection sums and symmetry Count routes using powers of a connection matrix Construct and analyse relation matrices