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

\(\displaystyle\to{B}+{A}={0}\)

\(\displaystyle\to{\left({B}+{A}\right)}-{A}={0}-{A}\)

\(\displaystyle\to{B}+{\left({A}-{A}\right)}=-{A}\)

\(\displaystyle\to{B}+{0}=-{A}\)

\(\displaystyle\to{B}=-{A}\)

\(\displaystyle\to{B}=-{\left[{1},{2},{4},{3}\right]}\)

PSK->B=[-1,-2,-4,-3]

Therefore, when A=[1,2,4,3], for B=[-1,-2,-4,-3], A+B=0.

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

\(\displaystyle\to{B}+{A}={0}\)

\(\displaystyle\to{\left({B}+{A}\right)}-{A}={0}-{A}\)

\(\displaystyle\to{B}+{\left({A}-{A}\right)}=-{A}\)

\(\displaystyle\to{B}+{0}=-{A}\)

\(\displaystyle\to{B}=-{A}\)

\(\displaystyle\to{B}=-{\left[{1},{2},{4},{3}\right]}\)

PSK->B=[-1,-2,-4,-3]

Therefore, when A=[1,2,4,3], for B=[-1,-2,-4,-3], A+B=0.