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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.

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It means that the first and third column are identical. – Darth Geek Jul 30 '14 at 13:08
up vote 1 down vote accepted

The first and last column have the same elements.

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


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

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

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so it is nothing to do with notation I can just replace all $a_{11} with say $x$ and so on if I understand you correctly – Gobabis Jul 30 '14 at 13:12
With what do you want to replace it? – evinda Jul 30 '14 at 13:16
Substituted $a_{11} , a_{12} , a_{13}$ with $x , y ,z$ and then worked out the determinant = 0 thank you – Gobabis Jul 30 '14 at 13:21
Nice!!!You are welcome!!! :) – evinda Jul 30 '14 at 13:22

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?

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Agreed that is what totally confused me and what you say makes more sense than the original question. thanx – Gobabis Jul 30 '14 at 13:26

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