Graph Theory Pretest Solutions

## Question 2 Solutions

### A)

2A is defined to be the matrix A where every entry is multiplied by the scalar 2. Hence we get that:

(1)
\begin{align} 2A = \begin{bmatrix} 4 & 6 & -2 & 0 & 10 & 4 & 18 \\ 8 & 12 & 4 & -14 & 22 & 2n & 0\\ 18 & 2k & -44 & 2 & 0 & 8 & 8\\ 18 & 6 & 4 & 0 & 4 & 8 & -2\\ -22 & 4 & 0 & 10 & 4n & 0 & 6 \end{bmatrix} \end{align}

### B)

The determinant of this matrix is not defined. The determinant is only defined for n x n (square) matrices.

### C)

We know that A is a 5x7 matrix. Hence matrix T must be a 7xr matrix for the matrix product AT to exist. Hence s = 7, and r can be any positive integer greater or equal to 1.

### D)

Note that $\sum_{k=1}^{n} a_{3n}$ is the sum of the entries in row 3. This sum is equal to k - 4.

### E)

Note that $\sum_{k=1}^{m} a_{6m}$ is the sum of the entries in column 6. This sum is equal to 10 + n.

### F)

A square matrix B is symmetric if the transpose of B is equal to be, that is $B = B^T$. Alternatively we could say that a matrix B is symmetric if all entries $b_{ij} = b_{ji}$ for every i, j.