# Find if possible the matrices: a) AB b) BA A=begin{bmatrix} -1 -2 -3 end{bmatrix} , B=begin{bmatrix}1 & 2 & 3 end{bmatrix}

Find if possible the matrices:
a) AB
b) BA
$A=\left[\begin{array}{c}-1\\ -2\\ -3\end{array}\right],B=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$
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broliY

Step 1
Given that:
The matrices,
$A=\left[\begin{array}{c}-1\\ -2\\ -3\end{array}\right],B=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$
Step 2
We know that,
Finding the product of two matrices is only possible when the inner dimension are same , meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix.
a) To find AB :
Let, $A=\left[\begin{array}{c}-1\\ -2\\ -3\end{array}\right],B=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$
Number of columns of first matrix (A) is 1 .
Number of rows of the second matrix is 1.
To get,
Step 3
Number of columns of $A=$ Number of rows of $B=1$.
Then,
$AB=\left[\begin{array}{c}-1\\ -2\\ -3\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\end{array}\right]=\left[\begin{array}{ccc}-1\left(1\right)& \left(-2\right)\left(2\right)& \left(-3\right)\left(3\right)\end{array}\right]=\left[\begin{array}{ccc}-1& -4& -9\end{array}\right]$
To get,
$AB=\left[\begin{array}{ccc}-1& -4& -9\end{array}\right]$
b) To find BA :
Number of columns of first matrix B is 3 and number of rows of matrix A is 3 which are equal .
Then,
Step 4
$BA=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]\left[\begin{array}{c}-1\\ -2\\ -3\end{array}\right]=\left[\begin{array}{c}1\left(-1\right)+2\left(-2\right)+3\left(-3\right)\end{array}\right]=\left[\begin{array}{c}-1-4-9\end{array}\right]=\left[\begin{array}{c}-14\end{array}\right]$
Therefore,
a) $AB=\left[\begin{array}{c}-1-4-9\end{array}\right]$
b) $BA=\left[\begin{array}{c}-14\end{array}\right]$

Jeffrey Jordon