Linear Algebra

a)Find c[Ta]c

The matrix A is =

[ 1  5  5]

[ 0  2  7]

[ 2  1 11]

. We find the matrix Ta=AT which is the transposed matrix of A

Dimensions of matrix A are 3 x 3 thus the size of the matrix Ta will also be 3 x 3.

We find the elements of the matrix Ta to be:

c1 1 = a1 1 = 1;

c2 1 = a1 2 = 5;

c3 1 = a1 3 = 5;

c1 2 = a2 1 = 0;

c2 2 = a2 2 = 2;

c3 2 = a2 3 = 7;

c1 3 = a3 1 = 2;

c2 3 = a3 2 = 1;

c3 3 = a3 3 = 11;

 

So, Ta =

[ 1  0  2]

[ 5  2  1]

[ 5  7 11]

This is the transpose, Ta.

Therefore c(Ta)c where c=   1   0   0

                                        0   1   0

                                        0   0   1

We first get the product of c and Ta after which we will then multiply by matrix c once more.

Product c*Ta will yield,        1  0  0      1   0   2

                                       0  1  0      5   2   1

                                       0  0  1      5   7   11

 

The first matrix is a 3*3 matrix and the second one is also a 3*3 thus the answer should also be a 3*3 matrix. Therefore multiplying the first row of the first row of the first matrix by the first column of the second matrix and so on, we get the product is

                 

                       (1*1)+(0*5)+(0*5)     (1*0)+(0*2)+(0*7)     (1*2)+(0*1)+(0*11)

                       (0*1)+(1*5)+(0*5)     (0*0)+(1*2)+(0*7      (0*2)+(1*1)+(0*11)

                       (0*1)+(0*5)+(1*5)     (0*0)+(0*2)+(1*7)      (0*2)+(0*1)+(1*11)

 This will be equal to,  1  0    2         multiplying this by c= 1  0  0 we get the final answer as

                                     5   2   1                                             0  1   0

                                     5   7  11                                            0  0   1

                       (1*1)+(0*5)+(0*5)     (1*0)+(0*2)+(0*7)     (1*2)+(0*1)+(0*11)

                       (0*1)+(1*5)+(0*5)     (0*0)+(1*2)+(0*7      (0*2)+(1*1)+(0*11)

                       (0*1)+(0*5)+(1*5)     (0*0)+(0*2)+(1*7)      (0*2)+(0*1)+(1*11)

 Thereforec(Ta)c will be, 1  0   2

                                          5   2   1

                                          5   7   11

b) Find c[Ta]b

 Ta is the transpose of A which is obtained as

The matrix A is =

[ 1  5  5]

[ 0  2  7]

[ 2  1 11]

. We find the matrix C=AT which is the transposed matrix A

Dimensions of matrix A are 3 x 3 thus the size of the matrix Ta is also 3 x 3.

We find the elements of the matrix Ta to be:

c1 1 = a1 1 = 1;

c2 1 = a1 2 = 5;

c3 1 = a1 3 = 5;

c1 2 = a2 1 = 0;

c2 2 = a2 2 = 2;

c3 2 = a2 3 = 7;

c1 3 = a3 1 = 2;

c2 3 = a3 2 = 1;

c3 3 = a3 3 = 11;

 

 

 

So, Ta =

 

[ 1  0  2]

[ 5  2  1]

[ 5  7 11]

This is the transpose, Ta.

The product c[Ta]b will be obtained by:

Multiplying c by Ta first we get, we know that c=  1   0   0   and Ta= 1   0   2

                                                                                  0   1   0                 5   2   1

                                                                                  0   0   1                 5   7   11

This product will be:

The first matrix is a 3*3 matrix and the second one is also a 3*3 thus the answer should also be a 3*3 matrix. Therefore multiplying the first row of the first matrix by the first column of the second matrix and so on, we get the product will be

                       (1*1)+(0*5)+(0*5)     (1*0)+(0*2)+(0*7)     (1*2)+(0*1)+(0*11)

                       (0*1)+(1*5)+(0*5)     (0*0)+(1*2)+(0*7      (0*2)+(1*1)+(0*11)

                       (0*1)+(0*5)+(1*5)     (0*0)+(0*2)+(1*7)      (0*2)+(0*1)+(1*11)

        =       1  0   2   then multiplying this by matrix b=  2   1   1

                  5   2   1                                                           1  2   1

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                  5   7   11                                                         1   1   2

 

The product yields: 

(2*1)+(1*5)+(1*5)    (2*0)+(1*2)+(1*7)   (2*2)+(1*1)+(1*11)

(1*1)+(2*5)+(1*5)     (1*0)+(2*2)+(1*7)   (1*2)+(2*1)+(1*11)

(1*1)+(1*5)+(2*5)      (1*0)+(1*2)+(2*7)   (1*2)+(1*1)+(2*11)

 Therefore c[Ta]b will be;   12    9     16

                                       16   11    15

                                       16    16   25

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