Answer to Question #305274 in Linear Algebra for Nhlonipho Manganyi

Question

Let X =

1 2

3 4

; E =

a

b



Find each of the following.

If the operation cannot be done : state undefined operation.

a) XE

b) EX

c) XT X where XT

stands for the transpose of X

Answer 1

Solution

Given that

$X=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ , $E=\begin{bmatrix} a \\ b \end{bmatrix}$, and $X^T=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^T=\begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$

Then

(a)

$XE=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} a \\ b \end{bmatrix}$

$XE=\begin{bmatrix} 1\times a+ 2\times b \\ 3\times a+ 4\times b \end{bmatrix}$

$XE=\begin{bmatrix} 2+2b \\ 3+4b \end{bmatrix}$

(b)

For $EX=\begin{bmatrix} a \\ b \end{bmatrix}\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

We can see that number of columns in the first matrix $E$ is just one and the number of rows in the second column $X$ is two. Hence this multiplication is not possible.

(c)

For,

$X^TX=\begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

$X^TX=\begin{bmatrix} 1+9 & 2+12 \\ 2+12 & 4+16 \end{bmatrix}$

$X^TX=\begin{bmatrix} 10 & 14 \\ 14 & 20 \end{bmatrix}$