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Proving $beginvmatrixa&b&c\b&c&a\c&a&bendvmatrix=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$

Calculate $beginVmatrix1&2\2&4endVmatrix$Compute the determinant $beginvmatrix sin x & cos x \ -sin y & cos y endvmatrix$ using the Sarrus' ruleProving a determinant equalityTo show: $beginvmatrix -bc & b^2+bc & c^2+bc\ a^2+ac & -ac & c^2+ac \ a^2+ab & b^2+ab & -ab endvmatrix= (ab+bc+ca)^3$Find the value of the determinant $scriptsize beginvmatrix 8-x & -10 & 6 \ 8 & -9-x & -1 \ 16 & -15 & 9-x endvmatrix$Given $beginvmatrix p & q & r \ s & t & u \ v & w & x endvmatrix = -3$, find $beginvmatrixp&2q&5r+4p\s&2t&5u+4s\v&2w&5x+4vendvmatrix$.$beginvmatrix 1 & a &bc \ 1& b & ac\ 1&c & ab endvmatrix=beginvmatrix 1 & a &a^2 \ 1& b&b^2 \ 1& b & c^2 endvmatrix$Prove that $beginvmatrix xa&yb&zc\ yc&za&xb\ zb&xc&ya\ endvmatrix=xyzbeginvmatrix a&b&c\ c&a&b\ b&c&a\ endvmatrix$ if $x+y+z=0$Prove the equality: $beginvmatrix -2a &a+b &a+c \ b+a&-2b &b+c \ c+a&c+b &-2c endvmatrix = 4(a+b)(b+c)(c+a)$Determinant of beginvmatrix -2a &a+b &a+c \ b+a& -2b &b+c \ c+a&c+b & -2c endvmatrix

0

$begingroup$

Prove:$$beginvmatrixa&b&c\b&c&a\c&a&bendvmatrix=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$

I tried to use the Laplace expansion, but it seems useless.

matrices determinant

share|cite|improve this question

edited Mar 17 at 12:54

Rodrigo de Azevedo

13.2k41960

asked Mar 17 at 12:06

DavidDavid

644

$endgroup$

add a comment | 

0

$begingroup$

Prove:$$beginvmatrixa&b&c\b&c&a\c&a&bendvmatrix=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$

I tried to use the Laplace expansion, but it seems useless.

matrices determinant

share|cite|improve this question

edited Mar 17 at 12:54

Rodrigo de Azevedo

13.2k41960

asked Mar 17 at 12:06

DavidDavid

644

$endgroup$

add a comment | 

$begingroup$

Prove:$$beginvmatrixa&b&c\b&c&a\c&a&bendvmatrix=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$

I tried to use the Laplace expansion, but it seems useless.

matrices determinant

share|cite|improve this question

edited Mar 17 at 12:54

Rodrigo de Azevedo

13.2k41960

asked Mar 17 at 12:06

DavidDavid

644

$endgroup$

share|cite|improve this question

edited Mar 17 at 12:54

Rodrigo de Azevedo

13.2k41960

asked Mar 17 at 12:06

DavidDavid

644

share|cite|improve this question

edited Mar 17 at 12:54

Rodrigo de Azevedo

13.2k41960

asked Mar 17 at 12:06

DavidDavid

644

edited Mar 17 at 12:54

Rodrigo de Azevedo

13.2k41960

edited Mar 17 at 12:54

Rodrigo de Azevedo

13.2k41960

asked Mar 17 at 12:06

DavidDavid

644

asked Mar 17 at 12:06

DavidDavid

644

1 Answer
1

active

oldest

votes

2

$begingroup$

Hint:

$C_1'=C_1+C_2+C_3$

$$beginvmatrixa&b&c\b&c&a\c&a&bendvmatrix$$

$$=beginvmatrixa+b+c&b&c\b+c+a&c&a\c+a+b&a&bendvmatrix$$

$$=(a+b+c)beginvmatrix1&b&c\1&c&a\1&a&bendvmatrix$$

Now either expand or

use $R_2'=R_2-R_1, R_3'=R_3-R_1$

share|cite|improve this answer

answered Mar 17 at 12:08

lab bhattacharjeelab bhattacharjee

227k15158278

$endgroup$

add a comment | 

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2

$begingroup$

Hint:

$C_1'=C_1+C_2+C_3$

$$beginvmatrixa&b&c\b&c&a\c&a&bendvmatrix$$

$$=beginvmatrixa+b+c&b&c\b+c+a&c&a\c+a+b&a&bendvmatrix$$

$$=(a+b+c)beginvmatrix1&b&c\1&c&a\1&a&bendvmatrix$$

Now either expand or

use $R_2'=R_2-R_1, R_3'=R_3-R_1$

share|cite|improve this answer

answered Mar 17 at 12:08

lab bhattacharjeelab bhattacharjee

227k15158278

$endgroup$

add a comment | 

2

$begingroup$

Hint:

$C_1'=C_1+C_2+C_3$

$$beginvmatrixa&b&c\b&c&a\c&a&bendvmatrix$$

$$=beginvmatrixa+b+c&b&c\b+c+a&c&a\c+a+b&a&bendvmatrix$$

$$=(a+b+c)beginvmatrix1&b&c\1&c&a\1&a&bendvmatrix$$

Now either expand or

use $R_2'=R_2-R_1, R_3'=R_3-R_1$

share|cite|improve this answer

answered Mar 17 at 12:08

lab bhattacharjeelab bhattacharjee

227k15158278

$endgroup$

add a comment | 

$begingroup$

Hint:

$C_1'=C_1+C_2+C_3$

$$beginvmatrixa&b&c\b&c&a\c&a&bendvmatrix$$

$$=beginvmatrixa+b+c&b&c\b+c+a&c&a\c+a+b&a&bendvmatrix$$

$$=(a+b+c)beginvmatrix1&b&c\1&c&a\1&a&bendvmatrix$$

Now either expand or

use $R_2'=R_2-R_1, R_3'=R_3-R_1$

share|cite|improve this answer

answered Mar 17 at 12:08

lab bhattacharjeelab bhattacharjee

227k15158278

$endgroup$

share|cite|improve this answer

answered Mar 17 at 12:08

lab bhattacharjeelab bhattacharjee

227k15158278

answered Mar 17 at 12:08

lab bhattacharjeelab bhattacharjee

227k15158278

answered Mar 17 at 12:08

lab bhattacharjeelab bhattacharjee

227k15158278