Find determinant and trace of product of non square matrices
Let $B in M_{3,2}(mathbb{R})$,$C in M_{2,3}(mathbb{R})$ so that $BC=begin{pmatrix}2 & -2 & 3\
0 & 0 & 3\
0 & 0 & 3
end{pmatrix}$. Find $det(CB)$ and $Tr(CB)$.
My attempt : I tried to compute the powers of $BC$ (I actually hoped that $BC$ was idempotent) and I saw that
$(BC)^n=begin{pmatrix}2^n & -2^n & 3^n\
0 & 0 & 3^n\
0 & 0 & 3^n
end{pmatrix}$ which doesn't really help.
linear-algebra matrices
asked 2 hours ago
Math Guy
405
add a comment |
Let $B in M_{3,2}(mathbb{R})$,$C in M_{2,3}(mathbb{R})$ so that $BC=begin{pmatrix}2 & -2 & 3\
0 & 0 & 3\
0 & 0 & 3
end{pmatrix}$. Find $det(CB)$ and $Tr(CB)$.
My attempt : I tried to compute the powers of $BC$ (I actually hoped that $BC$ was idempotent) and I saw that
$(BC)^n=begin{pmatrix}2^n & -2^n & 3^n\
0 & 0 & 3^n\
0 & 0 & 3^n
end{pmatrix}$ which doesn't really help.
linear-algebra matrices
asked 2 hours ago
Math Guy
405
@PeterMelech $det(C)$ doesn't exist because are not squared matrix
– Martín Vacas Vignolo
1 hour ago
Yes sorry, I noticed already
– Peter Melech
1 hour ago
@Yanko the trace formula doesn't hold for any non square matrices even though in this case according to the answer key it does.
– Math Guy
1 hour ago
@MathGuy Ok I believe you, anyway I post an answer without using this.
– Yanko
1 hour ago
add a comment |
Let $B in M_{3,2}(mathbb{R})$,$C in M_{2,3}(mathbb{R})$ so that $BC=begin{pmatrix}2 & -2 & 3\
0 & 0 & 3\
0 & 0 & 3
end{pmatrix}$. Find $det(CB)$ and $Tr(CB)$.
My attempt : I tried to compute the powers of $BC$ (I actually hoped that $BC$ was idempotent) and I saw that
$(BC)^n=begin{pmatrix}2^n & -2^n & 3^n\
0 & 0 & 3^n\
0 & 0 & 3^n
end{pmatrix}$ which doesn't really help.
linear-algebra matrices
asked 2 hours ago
Math Guy
405
Let $B in M_{3,2}(mathbb{R})$,$C in M_{2,3}(mathbb{R})$ so that $BC=begin{pmatrix}2 & -2 & 3\
0 & 0 & 3\
0 & 0 & 3
end{pmatrix}$. Find $det(CB)$ and $Tr(CB)$.
My attempt : I tried to compute the powers of $BC$ (I actually hoped that $BC$ was idempotent) and I saw that
$(BC)^n=begin{pmatrix}2^n & -2^n & 3^n\
0 & 0 & 3^n\
0 & 0 & 3^n
end{pmatrix}$ which doesn't really help.
linear-algebra matrices
linear-algebra matrices
asked 2 hours ago
Math Guy
405
asked 2 hours ago
Math Guy
405
edited 1 hour ago
asked 2 hours ago
Math Guy
405
asked 2 hours ago
Math Guy
405
asked 2 hours ago
Math Guy
405
405
@PeterMelech $det(C)$ doesn't exist because are not squared matrix
– Martín Vacas Vignolo
1 hour ago
Yes sorry, I noticed already
– Peter Melech
1 hour ago
@Yanko the trace formula doesn't hold for any non square matrices even though in this case according to the answer key it does.
– Math Guy
1 hour ago
@MathGuy Ok I believe you, anyway I post an answer without using this.
– Yanko
1 hour ago
add a comment |
@PeterMelech $det(C)$ doesn't exist because are not squared matrix
– Martín Vacas Vignolo
1 hour ago
Yes sorry, I noticed already
– Peter Melech
1 hour ago
@Yanko the trace formula doesn't hold for any non square matrices even though in this case according to the answer key it does.
– Math Guy
1 hour ago
@MathGuy Ok I believe you, anyway I post an answer without using this.
– Yanko
1 hour ago
@PeterMelech $det(C)$ doesn't exist because are not squared matrix
– Martín Vacas Vignolo
1 hour ago
@PeterMelech $det(C)$ doesn't exist because are not squared matrix
– Martín Vacas Vignolo
1 hour ago
Yes sorry, I noticed already
– Peter Melech
1 hour ago
Yes sorry, I noticed already
– Peter Melech
1 hour ago
@Yanko the trace formula doesn't hold for any non square matrices even though in this case according to the answer key it does.
– Math Guy
1 hour ago
@Yanko the trace formula doesn't hold for any non square matrices even though in this case according to the answer key it does.
– Math Guy
1 hour ago
@MathGuy Ok I believe you, anyway I post an answer without using this.
– Yanko
1 hour ago
@MathGuy Ok I believe you, anyway I post an answer without using this.
– Yanko
1 hour ago
add a comment |
1 Answer
1
active
oldest
votes
You need to work with this:
The trace of a matrix is the sum of the eigenvalues and the determinant is the product.
It is not hard to see that the eigenvalues of $BC$ are $2,3,0$. Let $v_1,v_2$ be the eigenvectors correspond to $2,3$ (you will see why we can't use the eigenvector for $0$ soon).
Consider the matrix $CB$ and look at the vectors $Cv_1,Cv_2$. Then $$CB(Cv_i) = C(BCv_i)=lambda_i Cv_i$$ therefore since $Cv_inot = 0$ (otherwise $BCv_i=0$) we have that $2,3$ are eigenvalues of $CB$ which means that the determinant is $6$ and the trace is $5$.
answered 1 hour ago
Yanko
6,029724
1
Excellent solution,congratulations !
– Math Guy
1 hour ago
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
You need to work with this:
The trace of a matrix is the sum of the eigenvalues and the determinant is the product.
It is not hard to see that the eigenvalues of $BC$ are $2,3,0$. Let $v_1,v_2$ be the eigenvectors correspond to $2,3$ (you will see why we can't use the eigenvector for $0$ soon).
Consider the matrix $CB$ and look at the vectors $Cv_1,Cv_2$. Then $$CB(Cv_i) = C(BCv_i)=lambda_i Cv_i$$ therefore since $Cv_inot = 0$ (otherwise $BCv_i=0$) we have that $2,3$ are eigenvalues of $CB$ which means that the determinant is $6$ and the trace is $5$.
answered 1 hour ago
Yanko
6,029724
1
Excellent solution,congratulations !
– Math Guy
1 hour ago
add a comment |
You need to work with this:
The trace of a matrix is the sum of the eigenvalues and the determinant is the product.
It is not hard to see that the eigenvalues of $BC$ are $2,3,0$. Let $v_1,v_2$ be the eigenvectors correspond to $2,3$ (you will see why we can't use the eigenvector for $0$ soon).
Consider the matrix $CB$ and look at the vectors $Cv_1,Cv_2$. Then $$CB(Cv_i) = C(BCv_i)=lambda_i Cv_i$$ therefore since $Cv_inot = 0$ (otherwise $BCv_i=0$) we have that $2,3$ are eigenvalues of $CB$ which means that the determinant is $6$ and the trace is $5$.
answered 1 hour ago
Yanko
6,029724
1
Excellent solution,congratulations !
– Math Guy
1 hour ago
add a comment |
You need to work with this:
The trace of a matrix is the sum of the eigenvalues and the determinant is the product.
It is not hard to see that the eigenvalues of $BC$ are $2,3,0$. Let $v_1,v_2$ be the eigenvectors correspond to $2,3$ (you will see why we can't use the eigenvector for $0$ soon).
Consider the matrix $CB$ and look at the vectors $Cv_1,Cv_2$. Then $$CB(Cv_i) = C(BCv_i)=lambda_i Cv_i$$ therefore since $Cv_inot = 0$ (otherwise $BCv_i=0$) we have that $2,3$ are eigenvalues of $CB$ which means that the determinant is $6$ and the trace is $5$.
answered 1 hour ago
Yanko
6,029724
You need to work with this:
The trace of a matrix is the sum of the eigenvalues and the determinant is the product.
It is not hard to see that the eigenvalues of $BC$ are $2,3,0$. Let $v_1,v_2$ be the eigenvectors correspond to $2,3$ (you will see why we can't use the eigenvector for $0$ soon).
Consider the matrix $CB$ and look at the vectors $Cv_1,Cv_2$. Then $$CB(Cv_i) = C(BCv_i)=lambda_i Cv_i$$ therefore since $Cv_inot = 0$ (otherwise $BCv_i=0$) we have that $2,3$ are eigenvalues of $CB$ which means that the determinant is $6$ and the trace is $5$.
answered 1 hour ago
Yanko
6,029724
answered 1 hour ago
Yanko
6,029724
answered 1 hour ago
Yanko
6,029724
answered 1 hour ago
Yanko
6,029724
6,029724
1
Excellent solution,congratulations !
– Math Guy
1 hour ago
add a comment |
1
Excellent solution,congratulations !
– Math Guy
1 hour ago
1
1
Excellent solution,congratulations !
– Math Guy
1 hour ago
Excellent solution,congratulations !
– Math Guy
1 hour ago
add a comment |
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@PeterMelech $det(C)$ doesn't exist because are not squared matrix
– Martín Vacas Vignolo
1 hour ago
Yes sorry, I noticed already
– Peter Melech
1 hour ago
@Yanko the trace formula doesn't hold for any non square matrices even though in this case according to the answer key it does.
– Math Guy
1 hour ago
@MathGuy Ok I believe you, anyway I post an answer without using this.
– Yanko
1 hour ago