Answer to Question #203634 in Linear Algebra for Om kaith

Question

If matrices

A =

1 5 3

2 5 7

B=

1

3

and C = [1 2].

compute A^tB, AC B^tA + CB, whenever defined. If you think any of these are not defined, give your reasons for saying so.

Answer 1

#$A^T=\begin{pmatrix} 1 & 2 \\ 5 & 5 \\ 3 & 7 \end{pmatrix}$

$B=\begin{pmatrix} 1 \\ 3 \\ \end{pmatrix}$

$A^TB=\begin{pmatrix} 1+6 \\ 5 +15 \\ 3 +21 \end{pmatrix} = \begin{pmatrix} 7 \\ 20 \\ 24 \end{pmatrix}$

So it exists

# Product of AC will no exist.As In order for matrix multiplication to be defined, the number of columns in the first matrix must be equal to the number of rows in the second matrix and which is not satisfied here.

# $B^T=[1 ,3]$

$A=$ $\begin{pmatrix} 1 & 5 & 3 \\ 2 & 5 &7 \\ \end{pmatrix}$

$B^T A= [1+6,5+15,3+21]=[7,20,24]$

$CB=[1+6]=[7]$

Now the order of $B^TA=1 \times3$

the order of $CB=1 \times 1$

Both orders are not same so we can not add both of them.