Answer to Question #117795 in Linear Algebra for mike

Question

if matrix A =
{5 3 -2}
{4 2 1}
{7 -1 4}
a) Multiply A by the matrix
1
1
1
(i)Explain the effect on the elements of A by this product of matrices?

b) (i) Multiply 1 1 1 by A
(ii) Explain the effect on the elements of A by this product of matrices?

Answer 1

Given,

$$A=\begin{bmatrix} 5 & 3&-2 \\ 4 & 2&1\\ 7&-1&4 \end{bmatrix}$$

a). We have to multiply to right by a column matrix

$$B=\begin{bmatrix} 1 \\ 1\\1 \end{bmatrix}$$

with $A$ ,thus

$$\begin{bmatrix} 5 & 3&-2 \\ 4 & 2&1\\ 7&-1&4 \end{bmatrix}\begin{bmatrix} 1 \\ 1\\1 \end{bmatrix}=\begin{bmatrix} 5\cdot1+3\cdot1+(-2)\cdot1\\ 4\cdot1+2\cdot1+1\cdot1\\ 7\cdot1+ (-1)\cdot1+4\cdot1 \end{bmatrix}=\begin{bmatrix} 6\\7\\10 \end{bmatrix}$$

On multiplying $B$ with$A$ , the elements of $A$ get multiplied by 1(multiplication of $B$ is done by each rows of $A$) thus there is no effect on the elements of $A$ but the final result get changed completely as before multiplication $A$ was $3\times3$ matrix but after multiplication its a $3\times 1$ column matrix.

b).Now we have to multiply transpose of $B$ i.e $B^t$ to left of $A$ ,thus

$$\begin{bmatrix} 1 & 1&1 \end{bmatrix}\begin{bmatrix} 5 & 3&-2 \\ 4 & 2&1\\ 7&-1&4 \end{bmatrix}=\\\begin{bmatrix} 1\cdot 5+1\cdot4+1\cdot7&1\cdot3+1\cdot2+1\cdot(-1)&1\cdot(-2)+1\cdot1+1\cdot4 \end{bmatrix}=\\\begin{bmatrix} 16 & 4&3 \end{bmatrix}$$

Clearly, as part (a) ,the elements of $A$ get multiplied by 1(in fact this time multiplication is done by each column of $A$ by $B$) and the finally $A$ becomes a $1\times3$ row matrix.

Hence we are done.