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Found in: Page 35

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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

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 $\left(\begin{array}{ccc}9& -9& 18\\ 27& 36& 45\end{array}\right)$

See the step by step solution

## Step 1: 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,

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

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

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

## Step 3: Consider the matrix and perform the product operation.

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

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

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 $\stackrel{\to }{x}$ is 9

Perform the row column operation.

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