Question

a. Find$\left(\begin{array}{cc}2& 3\\ 4& 5\\ 6& 7\end{array}\right)\mathbf{+}\left(\begin{array}{cc}7& 5\\ 3& 1\\ 0& -1\end{array}\right)$

b. Find$\mathbf{9}\left(\begin{array}{ccc}1& -1& 2\\ 3& 4& 5\end{array}\right)$

Short answer

a. The value of $\left(\begin{array}{cc}2& 3\\ 4& 5\\ 6& 7\end{array}\right)+\left(\begin{array}{cc}7& 5\\ 3& 1\\ 0& -1\end{array}\right)$is $\left(\begin{array}{cc}9& 8\\ 7& 6\\ 6& 6\end{array}\right)$

b. The value of $9\left(\begin{array}{ccc}1& -1& 2\\ 3& 4& 5\end{array}\right)$is$\left(\begin{array}{ccc}9& -9& 18\\ 27& 36& 45\end{array}\right)$

Step-by-step solution

Consider the matrices and perform the addition operation.

If A and B are the matrices of same size, then, the sum of matrices is,

$\left[\begin{array}{ccc}{a}_{11}& \dots & a_{1m}\\ ⋮& & ⋮\\ a_{n1}& \dots & anm\end{array}\right]+\left[\begin{array}{ccc}{b}_{11}& \dots & b_{1m}\\ ⋮& & ⋮\\ b_{n1}& \dots & bnm\end{array}\right]=\left[\begin{array}{cccc}{a}_{11}& +b_{11}& \dots a_{1m}& +b_{1m}\\ ⋮& ⋮& & \\ a_{n1}& +b_{n1}& \dots a_{nm}& b_{nm}\end{array}\right]$

The given matrices are,

$\left(\begin{array}{cc}2& 3\\ 4& 5\\ 6& 7\end{array}\right)+\left(\begin{array}{cc}7& 5\\ 3& 1\\ 0& -1\end{array}\right)$

Perform the addition operation.

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

$=\left(\begin{array}{cc}9& 8\\ 7& 6\\ 6& 6\end{array}\right)$

Consider the matrix and perform the product operation.

If A is an $n\times m$matrix with row vectors $\overrightarrow{\omega 1},......,\overrightarrow{\omega n}$and $\overrightarrow{x}$is a vector in ${\mathrm{ℝ}}^m$then,

$A\overrightarrow{x}=\left[\begin{array}{c}-\overrightarrow{\omega 1}-\\ ⋮\\ -\overrightarrow{\omega n}-\end{array}\right]\overrightarrow{x}=\left[\begin{array}{c}-\overrightarrow{\omega 1}.\overrightarrow{x}-\\ ⋮\\ -\overrightarrow{\omega n}.\overrightarrow{x}-\end{array}\right]$

The given matrix is,

$9\left(\begin{array}{ccc}1& -1& 2\\ 3& 4& 5\end{array}\right)$

Here A is $\left(\begin{array}{ccc}1& -1& 2\\ 3& 4& 5\end{array}\right)$and $\overrightarrow{x}$is 9

Perform the row column operation.

localid="1659341013254" $\begin{array}{l}=\left(\begin{array}{ccc}9.1& 9\left(-1\right)& 9.2\\ 9.3& 9.4& 9.5\end{array}\right)\\ =\left(\begin{array}{ccc}9& -9& 18\\ 27& 36& 45\end{array}\right)\end{array}$