# Recent questions tagged solution

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Say you have $k$ linear algebraic equations in $n$ variables; in matrix form we write $A X=Y$. Give a proof or counterexample for each of the following.
Say you have $k$ linear algebraic equations in $n$ variables; in matrix form we write $A X=Y$. Give a proof or counterexample for each of the following.Say you have $k$ linear algebraic equations in $n$ variables; in matrix form we write $A X=Y$. Give a proof or counterexample for each of the fo ...
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Solve the given system - or show that no solution exists: \begin{aligned} x+2 y &=& 1 \\ 3 x+2 y+& 4 z=& 7 \\ -2 x+y-& 2 z=&-1 \end{aligned}
Solve the given system - or show that no solution exists: \begin{aligned} x+2 y &=& 1 \\ 3 x+2 y+& 4 z=& 7 \\ -2 x+y-& 2 z=&-1 \end{aligned}Solve the given system - or show that no solution exists: \ \begin{aligned} x+2 y &amp;=&amp; 1 \\ 3 x+2 y+&amp; 4 z=&amp; 7 \\ -2 x+y-&amp; 2 z=&amp ...
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Consider the system of equations \begin{aligned} &x+y-z=a \\ &x-y+2 z=b . \end{aligned}
Consider the system of equations \begin{aligned} &x+y-z=a \\ &x-y+2 z=b . \end{aligned}Consider the system of equations \ \begin{aligned} &amp;x+y-z=a \\ &amp;x-y+2 z=b . \end{aligned} \ a) Find the general solution of the homogeneous ...
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If $A$ is a $5 \times 5$ matrix with $\operatorname{det} A=-1$, compute $\operatorname{det}(-2 A)$.
If $A$ is a $5 \times 5$ matrix with $\operatorname{det} A=-1$, compute $\operatorname{det}(-2 A)$.If $A$ is a $5 \times 5$ matrix with $\operatorname{det} A=-1$, compute $\operatorname{det}(-2 A)$. ...
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Identify all possible eigenvalues of an $n \times n$ matrix $A$ that satisfies the matrix equation: $A-2 I=-A^{2}$. Justify your answer.
Identify all possible eigenvalues of an $n \times n$ matrix $A$ that satisfies the matrix equation: $A-2 I=-A^{2}$. Justify your answer.a) Identify all possible eigenvalues of an $n \times n$ matrix $A$ that satisfies the matrix equation: $A-2 I=-A^{2}$. Justify your answer. b) M ...
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Let $A:=\left(\begin{array}{rrr}4 & 4 & 4 \\ -2 & -3 & -6 \\ 1 & 3 & 6\end{array}\right)$. Compute
Let $A:=\left(\begin{array}{rrr}4 & 4 & 4 \\ -2 & -3 & -6 \\ 1 & 3 & 6\end{array}\right)$. ComputeLet $A:=\left(\begin{array}{rrr}4 &amp; 4 &amp; 4 \\ -2 &amp; -3 &amp; -6 \\ 1 &amp; 3 &amp; 6\end{array}\right)$. Compute a) the characteristic po ...
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Compute the value of the determinant of the $3 \times 3$ complex matrix $X$, provided that...
Compute the value of the determinant of the $3 \times 3$ complex matrix $X$, provided that...Compute the value of the determinant of the $3 \times 3$ complex matrix $X$, provided that $\operatorname{tr}(X)=1, \operatorname{tr}\left(X^{2}\ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(A$ be a $3 \times 3$ matrix with eigenvalues $\lambda_{1}, \lambda_{2}, \lambda_{3}$ ...
Let $A$ be a $3 \times 3$ matrix with eigenvalues $\lambda_{1}, \lambda_{2}, \lambda_{3}$ ...Let $A$ be a $3 \times 3$ matrix with eigenvalues $\lambda_{1}, \lambda_{2}, \lambda_{3}$ and corresponding linearly independent eigenvectors $... close 0 answers 9 views Let \(A$ and $B$ be $n \times n$ complex matrices that commute: $A B=B A$. If $\lambda$ is an eigenvalue of $A$, let $\mathcal{V}_{\lambda}$ be the subspace of all eigenvectors having this eigenvalue.Let $A$ and $B$ be $n \times n$ complex matrices that commute: $A B=B A$. If $\lambda$ is an eigenvalue of $A$, let $\mathcal{V}_{\lambda ... close 0 answers 10 views Let \(M$ be a $2 \times 2$ matrix with the property that the sum of each of the rows and also the sum of each of the columns is the same constant $c$. Which (if any) any of the vectorsLet $M$ be a $2 \times 2$ matrix with the property that the sum of each of the rows and also the sum of each of the columns is the same constant \ ...
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Let $A$ be a square matrix and $p(\lambda)$ any polynomial. If $\lambda$ is an eigenvalue of $A$, show that $p(\lambda)$ is an eigenvalue of the matrix $p(A)$ with the same eigenvector.Let $A$ be a square matrix and $p(\lambda)$ any polynomial. If $\lambda$ is an eigenvalue of $A$, show that $p(\lambda)$ is an eigenvalue of ...
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Suppose that $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda_{1}=-1, \lambda_{2}=0$ and $\lambda_{3}=1$, and corresponding eigenvectors
Suppose that $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda_{1}=-1, \lambda_{2}=0$ and $\lambda_{3}=1$, and corresponding eigenvectorsSuppose that $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda_{1}=-1, \lambda_{2}=0$ and $\lambda_{3}=1$, and corresponding eigenvectors ...
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Let $C$ be a $2 \times 2$ matrix of real numbers. Give a proof or counterexample to each of the following assertions:
Let $C$ be a $2 \times 2$ matrix of real numbers. Give a proof or counterexample to each of the following assertions:Let $C$ be a $2 \times 2$ matrix of real numbers. Give a proof or counterexample to each of the following assertions: a) $\operatorname{det}\left ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(L$ be an $n \times n$ matrix with real entries and let $\lambda$ be an eigenvalue of $L$. In the following list, identify all the assertions that are correct.
Let $L$ be an $n \times n$ matrix with real entries and let $\lambda$ be an eigenvalue of $L$. In the following list, identify all the assertions that are correct.Let $L$ be an $n \times n$ matrix with real entries and let $\lambda$ be an eigenvalue of $L$. In the following list, identify all the asserti ...
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Let $a, b, c, d$, and $e$ be real numbers. For each of the following matrices, find their eigenvalues, corresponding eigenvectors, and orthogonal matrices that diagonalize them.Let $a, b, c, d$, and $e$ be real numbers. For each of the following matrices, find their eigenvalues, corresponding eigenvectors, and orthogonal ...
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Find an orthogonal matrix $R$ that diagonalizes $A:=\left(\begin{array}{rrr}1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 2\end{array}\right)$
Find an orthogonal matrix $R$ that diagonalizes $A:=\left(\begin{array}{rrr}1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 2\end{array}\right)$Find an orthogonal matrix $R$ that diagonalizes $A:=\left(\begin{array}{rrr}1 &amp; -1 &amp; 0 \\ -1 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 2\end{array ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(A=\left(\begin{array}{cc}a & b-a \\ 0 & b\end{array}\right)$ - Diagonalize $A$.
Let $A=\left(\begin{array}{cc}a & b-a \\ 0 & b\end{array}\right)$ - Diagonalize $A$.Let $A=\left(\begin{array}{cc}a &amp; b-a \\ 0 &amp; b\end{array}\right)$ a) Diagonalize $A$. b) Use this to compute $A^{k}$ for any integer $... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Diagonalize the matrix $A=\left(\begin{array}{lll} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{array}\right)$ 0 answers 4 views Diagonalize the matrix $A=\left(\begin{array}{lll} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{array}\right)$Diagonalize the matrix \ A=\left(\begin{array}{lll} 1 &amp; 0 &amp; 2 \\ 0 &amp; 1 &amp; 0 \\ 2 &amp; 0 &amp; 1 \end{array}\right) \ by finding the ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(A=\left(\begin{array}{lll}1 & 1 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2\end{array}\right)$.
Let $A=\left(\begin{array}{lll}1 & 1 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2\end{array}\right)$.Let $A=\left(\begin{array}{lll}1 &amp; 1 &amp; 2 \\ 1 &amp; 1 &amp; 2 \\ 1 &amp; 1 &amp; 2\end{array}\right)$. a) What is the dimension of the image ...
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[Frobenius] Let $A, B$, and $C$ be matrices so that the products $A B$ and $B C$ are defined.
[Frobenius] Let $A, B$, and $C$ be matrices so that the products $A B$ and $B C$ are defined.Frobenius Let $A, B$, and $C$ be matrices so that the products $A B$ and $B C$ are defined. Use the obvious \ \operatorname{dim}\left(\left ...
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Let $L$ be a $2 \times 2$ matrix. For each of the following give a proof or counterexample.
Let $L$ be a $2 \times 2$ matrix. For each of the following give a proof or counterexample.Let $L$ be a $2 \times 2$ matrix. For each of the following give a proof or counterexample. a) If $L^{2}=0$ then $L=0$. b) If $L^{2}=L$ the ...
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Let $A, B$, and $C$ be $n \times n$ matrices.
Let $A, B$, and $C$ be $n \times n$ matrices.Let $A, B$, and $C$ be $n \times n$ matrices. a) If $A^{2}$ is invertible, show that $A$ is invertible. NOTE: You cannot naively use the f ...
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Consider the homogeneous linear system $A x=0$ where $A=\left(\begin{array}{rrrr} 1 & 3 & 0 & 1 \\ 1 & 3 & -2 & -2 \\ 0 & 0 & 2 & 3 \end{array}\right) \text {. }$
Consider the homogeneous linear system $A x=0$ where $A=\left(\begin{array}{rrrr} 1 & 3 & 0 & 1 \\ 1 & 3 & -2 & -2 \\ 0 & 0 & 2 & 3 \end{array}\right) \text {. }$Consider the homogeneous linear system $A x=0$ where \ A=\left(\begin{array}{rrrr} 1 &amp; 3 &amp; 0 &amp; 1 \\ 1 &amp; 3 &amp; -2 &amp; -2 \\ 0 &a ...
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Let $A$ be a square matrix with integer elements. For each of the following give a proof or counterexample.
Let $A$ be a square matrix with integer elements. For each of the following give a proof or counterexample.Let $A$ be a square matrix with integer elements. For each of the following give a proof or counterexample. a) If $\operatorname{det}(A)=\pm 1$, t ...