Find the determinant of a $3AA^t$

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I am trying to find the determinant of $3AA^t$ or $|3AA^t|$

where

$$A = beginbmatrix -1 & 2 \ 2 & 3endbmatrix$$

So I wind up getting:

$$|A| = -3 - 4 = -7 = A^t$$

so $$|3AA^t| = -3 * -7 * -7 = 147$$

Is this right?

My check fails though. I try to find $3AA^t$ and I get this:

$$3AA^t = 3 * beginbmatrix -1 & 2 \ 2 & 3 endbmatrix * beginbmatrix -1 & 2 \ 2 & 3 endbmatrix $$

$$ =beginbmatrix -3 & 6 \ 6 & 9 endbmatrix * beginbmatrix -1 & 2
\ 2 & 3 endbmatrix $$

$$= beginbmatrix -15 & 12 \ 12 & 39 endbmatrix $$

$$= 441$$

which does not $= 147$. Where did I go wrong?

edited Aug 31 at 6:45

Asaf Karagilaâ™¦

294k31410737

asked Aug 30 at 22:18

Jwan622

1,75711224

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up vote
2
down vote

favorite

I am trying to find the determinant of $3AA^t$ or $|3AA^t|$

where

$$A = beginbmatrix -1 & 2 \ 2 & 3endbmatrix$$

So I wind up getting:

$$|A| = -3 - 4 = -7 = A^t$$

so $$|3AA^t| = -3 * -7 * -7 = 147$$

Is this right?

My check fails though. I try to find $3AA^t$ and I get this:

$$3AA^t = 3 * beginbmatrix -1 & 2 \ 2 & 3 endbmatrix * beginbmatrix -1 & 2 \ 2 & 3 endbmatrix $$

$$ =beginbmatrix -3 & 6 \ 6 & 9 endbmatrix * beginbmatrix -1 & 2
\ 2 & 3 endbmatrix $$

$$= beginbmatrix -15 & 12 \ 12 & 39 endbmatrix $$

$$= 441$$

which does not $= 147$. Where did I go wrong?

edited Aug 31 at 6:45

Asaf Karagilaâ™¦

294k31410737

asked Aug 30 at 22:18

Jwan622

1,75711224

add a commentÂ |Â

up vote
2
down vote

favorite

I am trying to find the determinant of $3AA^t$ or $|3AA^t|$

where

$$A = beginbmatrix -1 & 2 \ 2 & 3endbmatrix$$

So I wind up getting:

$$|A| = -3 - 4 = -7 = A^t$$

so $$|3AA^t| = -3 * -7 * -7 = 147$$

Is this right?

My check fails though. I try to find $3AA^t$ and I get this:

$$3AA^t = 3 * beginbmatrix -1 & 2 \ 2 & 3 endbmatrix * beginbmatrix -1 & 2 \ 2 & 3 endbmatrix $$

$$ =beginbmatrix -3 & 6 \ 6 & 9 endbmatrix * beginbmatrix -1 & 2
\ 2 & 3 endbmatrix $$

$$= beginbmatrix -15 & 12 \ 12 & 39 endbmatrix $$

$$= 441$$

which does not $= 147$. Where did I go wrong?

edited Aug 31 at 6:45

Asaf Karagilaâ™¦

294k31410737

asked Aug 30 at 22:18

Jwan622

1,75711224

I am trying to find the determinant of $3AA^t$ or $|3AA^t|$

where

$$A = beginbmatrix -1 & 2 \ 2 & 3endbmatrix$$

So I wind up getting:

$$|A| = -3 - 4 = -7 = A^t$$

so $$|3AA^t| = -3 * -7 * -7 = 147$$

Is this right?

My check fails though. I try to find $3AA^t$ and I get this:

$$3AA^t = 3 * beginbmatrix -1 & 2 \ 2 & 3 endbmatrix * beginbmatrix -1 & 2 \ 2 & 3 endbmatrix $$

$$ =beginbmatrix -3 & 6 \ 6 & 9 endbmatrix * beginbmatrix -1 & 2
\ 2 & 3 endbmatrix $$

$$= beginbmatrix -15 & 12 \ 12 & 39 endbmatrix $$

$$= 441$$

which does not $= 147$. Where did I go wrong?

edited Aug 31 at 6:45

Asaf Karagilaâ™¦

294k31410737

asked Aug 30 at 22:18

Jwan622

1,75711224

edited Aug 31 at 6:45

Asaf Karagilaâ™¦

294k31410737

edited Aug 31 at 6:45

Asaf Karagilaâ™¦

294k31410737

edited Aug 31 at 6:45

Asaf Karagilaâ™¦

294k31410737

asked Aug 30 at 22:18

Jwan622

1,75711224

asked Aug 30 at 22:18

Jwan622

1,75711224

asked Aug 30 at 22:18

Jwan622

1,75711224

add a commentÂ |Â

3 Answers
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If $A$ is an $n times n$ matrix, $det(3A) = det(3I) det(A) = 3^n det(A)$, not $3 det(A)$.

answered Aug 30 at 22:21

Robert Israel

306k22201443

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up vote
5
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Because if $lambda$ is a scalar and $A$ is a $ntimes n$ matrix, then $det(lambda A)=lambda^ndet A$. That's why you only got $frac13$ of the right answer.

answered Aug 30 at 22:23

JosÃ© Carlos Santos

120k16101182

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up vote
1
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$3$ is a scalar and A is a $2ÃƒÂ—2$ matrix,then $det(3A)=3^2det(A)$.$$|3AA^t|=3^2ÃƒÂ—|AA^t|$$
$$|AA^t|=det left( beginbmatrix -1 & 2 \ 2 & 3 endbmatrix ÃƒÂ— beginbmatrix -1 & 2 \ 2 & 3 endbmatrix right)=detleft(beginbmatrix
5 & 4
\ 4 & 13endbmatrix right)=49$$ $|3AA^t|=9ÃƒÂ—49=441$.
Sorry if i do any mistake.

edited Aug 31 at 14:17

answered Aug 31 at 7:25

md emon

878

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3 Answers
3

active

oldest

votes

3 Answers
3

active

oldest

votes

up vote
9
down vote

accepted

If $A$ is an $n times n$ matrix, $det(3A) = det(3I) det(A) = 3^n det(A)$, not $3 det(A)$.

answered Aug 30 at 22:21

Robert Israel

306k22201443

add a commentÂ |Â

up vote
9
down vote

accepted

If $A$ is an $n times n$ matrix, $det(3A) = det(3I) det(A) = 3^n det(A)$, not $3 det(A)$.

answered Aug 30 at 22:21

Robert Israel

306k22201443

add a commentÂ |Â

up vote
9
down vote

accepted

If $A$ is an $n times n$ matrix, $det(3A) = det(3I) det(A) = 3^n det(A)$, not $3 det(A)$.

answered Aug 30 at 22:21

Robert Israel

306k22201443

If $A$ is an $n times n$ matrix, $det(3A) = det(3I) det(A) = 3^n det(A)$, not $3 det(A)$.

answered Aug 30 at 22:21

Robert Israel

306k22201443

answered Aug 30 at 22:21

Robert Israel

306k22201443

answered Aug 30 at 22:21

Robert Israel

306k22201443

answered Aug 30 at 22:21

Robert Israel

306k22201443

add a commentÂ |Â

up vote
5
down vote

Because if $lambda$ is a scalar and $A$ is a $ntimes n$ matrix, then $det(lambda A)=lambda^ndet A$. That's why you only got $frac13$ of the right answer.

answered Aug 30 at 22:23

120k16101182

add a commentÂ |Â

up vote
5
down vote

Because if $lambda$ is a scalar and $A$ is a $ntimes n$ matrix, then $det(lambda A)=lambda^ndet A$. That's why you only got $frac13$ of the right answer.

answered Aug 30 at 22:23

120k16101182

add a commentÂ |Â

up vote
5
down vote

Because if $lambda$ is a scalar and $A$ is a $ntimes n$ matrix, then $det(lambda A)=lambda^ndet A$. That's why you only got $frac13$ of the right answer.

answered Aug 30 at 22:23

120k16101182

Because if $lambda$ is a scalar and $A$ is a $ntimes n$ matrix, then $det(lambda A)=lambda^ndet A$. That's why you only got $frac13$ of the right answer.

answered Aug 30 at 22:23

120k16101182

answered Aug 30 at 22:23

120k16101182

answered Aug 30 at 22:23

120k16101182

answered Aug 30 at 22:23

120k16101182

add a commentÂ |Â

up vote
1
down vote

edited Aug 31 at 14:17

answered Aug 31 at 7:25

md emon

878

add a commentÂ |Â

up vote
1
down vote

edited Aug 31 at 14:17

answered Aug 31 at 7:25

md emon

878

add a commentÂ |Â

up vote
1
down vote

edited Aug 31 at 14:17

answered Aug 31 at 7:25

md emon

878

edited Aug 31 at 14:17

answered Aug 31 at 7:25

md emon

878

edited Aug 31 at 14:17

answered Aug 31 at 7:25

md emon

878

answered Aug 31 at 7:25

md emon

878

answered Aug 31 at 7:25

md emon

878

add a commentÂ |Â

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