Question

Write a matrix that, when added to a $3\times 2$matrix, increases each element in the matrix by 4.

Short answer

A matrixwhen added to a $3\times 2$matrix, increases each element in the matrix by 4 is $A=\left[\begin{array}{cc}4& 4\\ 4& 4\\ 4& 4\end{array}\right]$.

Step-by-step solution

- 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.

- Write the required matrix

In order to find a matrix that when added to a $3\times 2$ matrix, increases each element in the matrix by 4 note that since matrices are to be added therefore the order of both the matrices must be same. Consider a matrix of order $3\times 2$,$A=\left[\begin{array}{cc}4& 4\\ 4& 4\\ 4& 4\end{array}\right].$

- Verify

Add the matrices A to a matrix $B=\left[\begin{array}{cc}1& 0\\ 2& 3\\ 5& -1\end{array}\right]$chosen at random. So as to verify if $A$when added to B increases each element in the matrix by 4, add the corresponding elements of both the matrices.

$\begin{array}{c}A+B=\left[\begin{array}{cc}4& 4\\ 4& 4\\ 4& 4\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 2& 3\\ 5& -1\end{array}\right]\\ =\left[\begin{array}{cc}4+1& 4+0\\ 4+2& 4+3\\ 4+5& 4+\left(-1\right)\end{array}\right]\\ =\left[\begin{array}{cc}5& 4\\ 6& 7\\ 9& 3\end{array}\right]\end{array}$