Q2. - Algebra 2

Question

Give an example of two matrices whose sum is zero matrix.

Step-by-Step Solution

Answer

Two matrices whose sum is a zero matrix are $A=\left[\begin{array}{cc}2& -5\\ -1& 3\end{array}\right]$ and $B=\left[\begin{array}{cc}-2& 5\\ 1& -3\end{array}\right]$. 

Step 1 - Define concept of matrix addition

If A and B are two matrices of order $m\times n$, then the addition of the two matrices will also yield a matrix of order $m\times n$ wherein each element is the sum of the corresponding elements.

Step 2 - Write two matrices

Write two matrices A and B such that the sum of the two matrices is zero. Note that since the sum has to be zero the order of both the matrices must be same. Consider the matrices of order $2\times 2$.

Let $A=\left[\begin{array}{cc}2& -5\\ -1& 3\end{array}\right]$ and $B=\left[\begin{array}{cc}-2& 5\\ 1& -3\end{array}\right]$.

Step 3 - Verify

Add the matrices A and B so as to verify if $A+B$ is a zero matrix. A zero matrix is a matrix whose each element is zero.

$\begin{array}{c}A+B=\left[\begin{array}{cc}2& -5\\ -1& 3\end{array}\right]+\left[\begin{array}{cc}-2& 5\\ 1& -3\end{array}\right]\\ =\left[\begin{array}{cc}2+\left(-2\right)& -5+5\\ -1+1& 3+\left(-3\right)\end{array}\right]\\ =\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\end{array}$