Matrix notation why is column 3= column 1?

Question

let $A =$\begin{bmatrix}a_{11} & a_{21} & a_{11}\\a_{12} & a_{22} & a_{12}\\a_{13} & a_{23} & a_{13}\end{bmatrix}

where $a_{ij}\in\Bbb R$ for each $1\le i , j\le 3$ which of the following is/are true

A. det(A)=0

B. A is invertible

I am having trouble undersanding : $1\le i , j\le 3$ and why column 3s entries are identical to column 1, does it mean they are equal as this is not what I expect from normal matrix notation.

Answer 1

The first and last column have the same elements.

If you find the determinant,you will see that it is equal to $0$.

EDIT:

The general form of a $3 \times 3$ matrix is:

$$\begin{bmatrix} a_{11} &a_{12} &a_{13} \\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{bmatrix}$$

Answer 2

It's a badly written question. Assuming that your text has already said that $a_{ij}$ denotes the element in the $i$th row and $j$th column of matrix $A$, far better would be to say,

"Suppose that $A$ is a $3 \times 3$ matrix with the property that $a_{31} = a_{11}, a_{32} = a_{12},$ and $a_{33} = a_{13}$. Which of the following three statements must be true?"

Does that help?