Q3. - Algebra 2

Question

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

Step-by-Step Solution

Answer

A matrix when 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 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 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 $3\times 2$, $A=\left[\begin{array}{cc}4& 4\\ 4& 4\\ 4& 4\end{array}\right].$

Step 3 - 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.

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