Construct a 3 4 matrix A = [a_ ij ] whose element a_ ij are given by: a_ ij = i - j

Construct a 3 4 matrix A = [a_ ij ] whose element a_ ij are given…

Question

Construct a $3 \times 4$ matrix $A = [a_{ij}]$ whose element $a_{ij}$ are given by:
$a_{ij} = i - j$

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Answer

Here, $\text{A}=(\text{a}_\text{ij})_{3\times4}=\begin{bmatrix}\text{a}_{11}&\text{a}_{12}&\text{a}_{13}&\text{a}_{14}\\\text{a}_{21}&\text{a}_{22}&\text{a}_{23}&\text{a}_{24}\\\text{a}_{31}&\text{a}_{32}&\text{a}_{33}&\text{a}_{34}\end{bmatrix}\ \dots(1)$
$a_{11} = 1 - 1 = 0, a_{12} = 1 - 2 = -1, a_{13} = 1 - 3 = -2, a_{14} = 1 - 4 = -3$
$a_{21} = 2 - 1 = 1, a_{22} = 2 - 2 = 0, a_{23} = 2 - 3 = -1, a_{24} = 2 - 4 = -2$
$a_{31} = 3 - 1 = 2, a_{32} = 3 - 2 = 1, a_{33} = 3 - 3 = 0$ and $a_{34} = 3 - 4 = -1$
So, the required matrix is $\begin{bmatrix}0&-1&-2&-3\\1&0&-1&-2\\2&1&0&-1\end{bmatrix}.$

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