Answer to Question #117795 in Linear Algebra for mike

Question #117795
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?
1
Expert's answer
2020-05-25T20:00:42-0400

Given,


A=[532421714]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=[111]B=\begin{bmatrix} 1 \\ 1\\1 \end{bmatrix}

with AA ,thus

[532421714][111]=[51+31+(2)141+21+1171+(1)1+41]=[6710]\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 BB withAA , the elements of AA get multiplied by 1(multiplication of BB is done by each rows of AA) thus there is no effect on the elements of AA but the final result get changed completely as before multiplication AA was 3×33\times3 matrix but after multiplication its a 3×13\times 1 column matrix.


b).Now we have to multiply transpose of BB i.e BtB^t to left of AA ,thus

[111][532421714]=[15+14+1713+12+1(1)1(2)+11+14]=[1643]\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 AA get multiplied by 1(in fact this time multiplication is done by each column of AA by BB) and the finally AA becomes a 1×31\times3 row matrix.

Hence we are done.


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