Question

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}

Matrices
ANSWERED
asked 2021-01-02
Find if possible the matrices:
a) AB
b) BA
\(A=\begin{bmatrix} -1 \\ -2 \\ -3 \end{bmatrix} , B=\begin{bmatrix}1 & 2 & 3 \end{bmatrix}\)

Answers (1)

2021-01-03

Step 1
Given that:
The matrices,
\(A=\begin{bmatrix} -1 \\ -2 \\ -3 \end{bmatrix} , B=\begin{bmatrix}1 & 2 & 3 \end{bmatrix}\)
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, \(A=\begin{bmatrix} -1 \\ -2 \\ -3 \end{bmatrix} , B=\begin{bmatrix}1 & 2 & 3 \end{bmatrix}\)
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,
\(AB=\begin{bmatrix} -1 \\ -2 \\ -3 \end{bmatrix} \begin{bmatrix}1 & 2 & 3 \end{bmatrix} = \begin{bmatrix}-1(1) & (-2)(2) & (-3)(3) \end{bmatrix}=\begin{bmatrix}-1 & -4 & -9 \end{bmatrix}\)
To get,
\(AB=\begin{bmatrix}-1 & -4 & -9 \end{bmatrix}\)
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
\(BA=\begin{bmatrix}1 & 2 & 3 \end{bmatrix}\begin{bmatrix} -1 \\ -2 \\ -3 \end{bmatrix}=\begin{bmatrix}1(-1) + 2(-2) + 3(-3) \end{bmatrix}=\begin{bmatrix}-1-4-9 \end{bmatrix}=\begin{bmatrix}-14 \end{bmatrix}\)
Therefore,
a) \(AB =\begin{bmatrix}-1-4-9 \end{bmatrix}\)
b) \(BA =\begin{bmatrix} -14 \end{bmatrix}\)

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