Construct a $3 \times 3$ matrix whose elements $a_{i j}$ are given by $a_{i j} = 0$ if $i < j$ and $a_{i j} = i + j$ if $i \geq j$ .

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Solution

The elements of a $3 \times 3$ matrix are: $A = ⎡ ⎢ ⎣ \begin{matrix} a_{11} & a_{12} & a_{13} a_{21} & a_{22} & a_{23} a_{31} & a_{32} & a_{33} \end{matrix} ⎤ ⎥ ⎦$

Applying the given rule: $a_{i j} = 0$ if $i < j$ and $a_{i j} = i + j$ if $i \geq j$ .
$a_{11} = 2, a_{12} = 0, a_{13} = 0 a_{21} = 3, a_{22} = 4, a_{23} = 0 a_{31} = 4, a_{32} = 5, a_{33} = 6$

$A = ⎛ ⎜ ⎝ \begin{matrix} 2 & 0 & 0 3 & 4 & 6 4 & 5 & 6 \end{matrix} ⎞ ⎟ ⎠$ is the required matrix

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Construct a $3 \times 3$ matrix whose elements $a_{i j}$ are given by $a_{i j} = 0$ if $i \neq j$ and $a_{i j} = i \times j$ if $i = j$

\frac{1}{2} |- 3 i + j|

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