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Solve the given system - or show that no solution exists: \[ \begin{aligned} x+2 y &=& 1 \\ 3 x+2 y+& 4 z=& 7 \\ -2 x+y-& 2 z=&-1 \end{aligned} \]
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Consider the system of equations \[ \begin{aligned} &x+y-z=a \\ &x-y+2 z=b . \end{aligned} \]
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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} \ 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 \(\operatorname{det}(-2 A)\).
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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.

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Let \(A:=\left(\begin{array}{rrr}4 & 4 & 4 \\ -2 & -3 & -6 \\ 1 & 3 & 6\end{array}\right)\). Compute
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Let \(A:=\left(\begin{array}{rrr}4 & 4 & 4 \\ -2 & -3 & -6 \\ 1 & 3 & 6\end{array}\right)\). ComputeLet \(A:=\left(\begin{array}{rrr}4 & 4 & 4 \\ -2 & -3 & -6 \\ 1 & 3 & 6\end{array}\right)\). Compute a) the characteristic po ...
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asked in Mathematics Jan 21
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Compute the value of the determinant of the \(3 \times 3\) complex matrix \(X\), provided that...
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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 \(\operatorname{tr}(X)=1, \operatorname{tr}\left(X^{2}\ ...
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asked in Mathematics Jan 21
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Let \(A\) be a \(3 \times 3\) matrix with eigenvalues \(\lambda_{1}, \lambda_{2}, \lambda_{3}\) ...
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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 \( ...
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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

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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:
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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: a) \(\operatorname{det}\left ...
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asked in Mathematics Jan 21
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Find an orthogonal matrix \(R\) that diagonalizes \(A:=\left(\begin{array}{rrr}1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 2\end{array}\right)\)
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Let \(A=\left(\begin{array}{cc}a & b-a \\ 0 & b\end{array}\right)\) - Diagonalize \(A\).
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Let \(A=\left(\begin{array}{cc}a & b-a \\ 0 & b\end{array}\right)\) - Diagonalize \(A\).Let \(A=\left(\begin{array}{cc}a & b-a \\ 0 & b\end{array}\right)\) a) Diagonalize \(A\). b) Use this to compute \(A^{k}\) for any integer \( ...
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asked in Mathematics Jan 21
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Diagonalize the matrix \[ A=\left(\begin{array}{lll} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{array}\right) \]
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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 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{array}\right) \ by finding the ...
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asked in Mathematics Jan 21
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Let \(A=\left(\begin{array}{lll}1 & 1 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2\end{array}\right)\).
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Let \(A=\left(\begin{array}{lll}1 & 1 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2\end{array}\right)\).Let \(A=\left(\begin{array}{lll}1 & 1 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2\end{array}\right)\). a) What is the dimension of the image ...
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asked in Mathematics Jan 21
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[Frobenius] Let \(A, B\), and \(C\) be matrices so that the products \(A B\) and \(B C\) are defined.
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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. Use the obvious \ \operatorname{dim}\left(\left ...
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asked in Mathematics Jan 21
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Let \(L\) be a \(2 \times 2\) matrix. For each of the following give a proof or counterexample.
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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. a) If \(L^{2}=0\) then \(L=0\). b) If \(L^{2}=L\) the ...
by MathsGee Platinum
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asked in Mathematics Jan 21
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Let \(A, B\), and \(C\) be \(n \times n\) matrices.
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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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Let \(A\) be a square matrix with integer elements. For each of the following give a proof or counterexample.
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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. a) If \(\operatorname{det}(A)=\pm 1\), t ...
by MathsGee Platinum
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asked in Mathematics Jan 20
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Find the general solution of the homogeneous equation.
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Find the general solution of the homogeneous equation.Consider the system of equations \ \begin{aligned} &x+y-z=a \\ &x-y+2 z=b \\ &3 x+y=c \end{aligned} \ a) Find the general solution of th ...
by MathsGee Platinum
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asked in Mathematics Jan 20
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