Question

Assume A and B are matrices over $\mathrm{\mathbb{Z}}$. Find $\mathit{A}\mathbf{+}\mathit{B}$.

1. $\mathit{A}\mathbf{=}\mathbf{\left[}\begin{array}{cccc}\mathbf{1}& \mathbf{2}& \mathbf{-}\mathbf{2}& \mathbf{0}\\ \mathbf{3}& \mathbf{5}& \mathbf{7}& \mathbf{11}\end{array}\mathbf{\right]}$and$\mathit{B}\mathbf{=}\mathbf{\left[}\begin{array}{cccc}\mathbf{0}& \mathbf{-}\mathbf{8}& \mathbf{2}& \mathbf{4}\\ \mathbf{6}& \mathbf{0}& \mathbf{4}& \mathbf{1}\end{array}\mathbf{\right]}$
2. $\mathit{A}\mathbf{=}\mathbf{\left[}\begin{array}{ccc}\mathbf{3}& \mathbf{0}& \mathbf{2}\\ \mathbf{4}& \mathbf{1}& \mathbf{6}\\ \mathbf{0}& \mathbf{1}& \mathbf{0}\\ \mathbf{2}& \mathbf{-}\mathbf{5}& \mathbf{7}\end{array}\mathbf{\right]}$and$\mathit{B}\mathbf{=}\mathbf{\left[}\begin{array}{ccc}\mathbf{1}& \mathbf{-}\mathbf{2}& \mathbf{0}\\ \mathbf{3}& \mathbf{0}& \mathbf{4}\\ \mathbf{0}& \mathbf{7}& \mathbf{-}\mathbf{6}\\ \mathbf{1}& \mathbf{6}& \mathbf{0}\end{array}\mathbf{\right]}$

Short answer

A) The value of$A\mathbf{+}B$ is $\left[\begin{array}{cccc}1& -6& 0& 4\\ 9& 5& 11& 12\end{array}\right]$.

B) The value of $A\mathbf{+}B$is$\left[\begin{array}{ccc}4& -2& 2\\ 7& 1& 10\\ 0& 8& \mathbf{-}6\\ 3& 1& 7\end{array}\right]$ .

Step-by-step solution

Consider the given matrices and explain the method to find the sum (a)

The given matrices are $A=\left[\begin{array}{cccc}1& 2& -2& 0\\ 3& 5& 7& 11\end{array}\right]$and$B=\left[\begin{array}{cccc}0& -8& 2& 4\\ 6& 0& 4& 1\end{array}\right]$.

If A and B are $2\mathbf{\times }4$ matrices, then their matrix sum $A\mathbf{+}B$ is the $2\mathbf{\times }4$ matrix with $a_{2\mathbf{\times }4}\mathbf{+}{b}_{2\mathbf{\times }4}$ in position data-custom-editor="chemistry" $2\mathbf{\times }4$. Now add the entries in corresponding positions over$\mathbb{Z}$.

Add the matrices A and B 

Add both the matrices A and B.

$\begin{array}{c}A\mathbf{+}B=\left[\begin{array}{cccc}1& 2& -2& 0\\ 3& 5& 7& 11\end{array}\right]+\left[\begin{array}{cccc}0& -8& 2& 4\\ 6& 0& 4& 1\end{array}\right]\\ =\left[\begin{array}{cccc}1+0& 2-8& -2+2& 0+4\\ 3+6& 5+0& 7+4& 11+1\end{array}\right]\\ =\left[\begin{array}{cccc}1& -6& 0& 4\\ 9& 5& 11& 12\end{array}\right]\end{array}$

Consider the given matrices and explain the method to find the sum (b) 

The given matrices are $A=\left[\begin{array}{ccc}3& 0& 2\\ 4& 1& 6\\ 0& 1& 0\\ 2& -5& 7\end{array}\right]$and$B=\left[\begin{array}{ccc}1& -2& 0\\ 3& 0& 4\\ 0& 7& -6\\ 1& 6& 0\end{array}\right]$.

If A and B are $4\mathbf{\times }3$ matrices, then their matrix sum$A\mathbf{+}B$ is the $4\mathbf{\times }3$ matrix with $a_{4\mathbf{\times }3}\mathbf{+}{b}_{4\mathbf{\times }3}$ in position $4\mathbf{\times }3$. Now add the entries in corresponding positions over$\mathbb{Z}$

Add the matrices A and B 

Add both the matrices A and B.

$\begin{array}{c}A\mathbf{+}B=\left[\begin{array}{ccc}3& 0& 2\\ 4& 1& 6\\ 0& 1& 0\\ 2& -5& 7\end{array}\right]\mathbf{+}\left[\begin{array}{ccc}1& -2& 0\\ 3& 0& 4\\ 0& 7& -6\\ 1& 6& 0\end{array}\right]\\ =\left[\begin{array}{ccc}3\mathbf{+}1& 0-2& 2\mathbf{+}0\\ 4\mathbf{+}3& 1\mathbf{+}0& 6\mathbf{+}4\\ 0\mathbf{+}0& 1\mathbf{+}7& 0\mathbf{-}6\\ 2\mathbf{+}1& -5\mathbf{+}6& 7\mathbf{+}0\end{array}\right]\\ =\left[\begin{array}{ccc}4& -2& 2\\ 7& 1& 10\\ 0& 8& \mathbf{-}6\\ 3& 1& 7\end{array}\right]\end{array}$