# Let A=begin{bmatrix}2 & -1&5 -3 & 4&0 end{bmatrix} text{ and } B=begin{bmatrix}-3 & -4&2 -1 & 0&-5 end{bmatrix} Find each result. 3A+2B=? A-3B=?

Let
Find each result.
3A+2B=?
A-3B=?
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Layton
Step 1
We have to find 3A+2B and A-3B where

In matrix if we multiply by any scalar then it get multiply in all the elements of the matrix example:
$2\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}2a& 2b\\ 2c& 2d\end{array}\right]$
And the operation of matrices will be in corresponding entities of the matrices.
So finding 3A+2B,
$3A+2B=3\left[\begin{array}{ccc}2& -1& 5\\ -3& 4& 0\end{array}\right]+2\left[\begin{array}{ccc}-3& -4& 2\\ -1& 0& -5\end{array}\right]$
$=\left[\begin{array}{ccc}6& -3& 15\\ -9& 12& 0\end{array}\right]+\left[\begin{array}{ccc}-6& -8& 4\\ -2& 0& -10\end{array}\right]$
$=\left[\begin{array}{ccc}0& -11& 19\\ -11& 12& -10\end{array}\right]$
Hence, value of 3A+2B is $\left[\begin{array}{ccc}0& -11& 19\\ -11& 12& -10\end{array}\right]$
Step 2
Finding A−3B,
$A-3B=\left[\begin{array}{ccc}2& -1& 5\\ -3& 4& 0\end{array}\right]-3\left[\begin{array}{ccc}-3& -4& 2\\ -1& 0& -5\end{array}\right]$
$=\left[\begin{array}{ccc}2& -1& 5\\ -3& 4& 0\end{array}\right]+\left[\begin{array}{ccc}-9& -12& 6\\ -3& 0& -15\end{array}\right]$
$=\left[\begin{array}{ccc}2-\left(-9\right)& -1-\left(-12\right)& 5-6\\ -3-\left(-3\right)& 4-0& 0-\left(-15\right)\end{array}\right]$
$=\left[\begin{array}{ccc}2+9& -1+12& -1\\ -3+3& 4& 0+15\end{array}\right]$
$=\left[\begin{array}{ccc}11& 11& -1\\ 0& 4& 15\end{array}\right]$
Hence, value of A-3B is $\left[\begin{array}{ccc}11& 11& -1\\ 0& 4& 15\end{array}\right]$
Jeffrey Jordon