# If A =[1,2,4,3], find B such that A+B=0

Matrix transformations

If $$A =[1,2,4,3]$$, find B such that $$A+B=0$$

2021-02-15

Recall: Theorem: The matrix addition is associative that is let A,B,C be matrices of order M x n. Then, $$(A+B)+C=A+(B+C)$$.
Theorem: The matrix addition is commutative that is let A and B be matrices of order M x n. Then $$A+B=B+A$$.
The given matrix is, $$A=[1,2,4,3]$$.
We have to find a matrix B such that $$A+B=0$$, where 0 is the zero matrix. Now,
$$A+B=0$$
$$\displaystyle\to{B}+{A}={0}$$
$$\displaystyle\to{\left({B}+{A}\right)}-{A}={0}-{A}$$
$$\displaystyle\to{B}+{\left({A}-{A}\right)}=-{A}$$
$$\displaystyle\to{B}+{0}=-{A}$$
$$\displaystyle\to{B}=-{A}$$
$$\displaystyle\to{B}=-{\left[{1},{2},{4},{3}\right]}$$
$$\rightarrow B=[-1,-2,-4,-3]$$
Therefore, when $$A=[1,2,4,3]$$, for $$B=[-1,-2,-4,-3], A+B=0$$.
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