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An $n \times n$ matrix is called nilpotent if $A^{k}$ equals the zero matrix for some positive integer $k$. (For instance, $\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right)$ is nilpotent.)An $n \times n$ matrix is called nilpotent if $A^{k}$ equals the zero matrix for some positive integer $k$. (For instance, $\left(\begin{array} ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Show that \(\operatorname{trace}(A B C)=\operatorname{trace}(C A B)=\operatorname{trace}(B C A)$.
Show that $\operatorname{trace}(A B C)=\operatorname{trace}(C A B)=\operatorname{trace}(B C A)$.Let $A, B$, and $C$ be any $n \times n$ matrices. a) Show that $\operatorname{trace}(A B)=\operatorname{trace}(B A)$. b) Show that $\operator ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Find the dimension of \(A$ considered as a real vector space.
Find the dimension of $A$ considered as a real vector space.Consider the two linear transformations on the vector space $V=\mathbf{R}^{n}$ : $R=$ right shift: $\left(x_{1}, \ldots, x_{n}\right) \rightarrow ... close 0 answers 2 views Let \(A: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ and $B: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$, so $B A: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ and $A B: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$.Let $A: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ and $B: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$, so $B A: \mathbb{R}^{3} \rightarrow \mathb ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a real matrix, not necessarily square.
Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a real matrix, not necessarily square.Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a real matrix, not necessarily square. a) If two columns of $A$ are the same, show that $A ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a linear map. Show that the following are equivalent.
Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a linear map. Show that the following are equivalent.Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a linear map. Show that the following are equivalent. a) For every $y \in \mathbb{R}^{k}$ th ...
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Let $C(\mathbb{R})$ be the linear space of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$.
Let $C(\mathbb{R})$ be the linear space of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$.Let $C(\mathbb{R})$ be the linear space of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$. a) Let $S_{c}$ be the set of different ...
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The five-digit number 1234A is divisible by 9. What is the remainder when it is divided by 7 ?
The five-digit number 1234A is divisible by 9. What is the remainder when it is divided by 7 ?The five-digit number 1234A is divisible by 9. What is the remainder when it is divided by 7 ? ...
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Determine $\frac{\mathrm{d} y}{\mathrm{~d} x}$ if $y=\left(x^{2}+\frac{1}{x^{2}}\right)^{2}$
Determine $\frac{\mathrm{d} y}{\mathrm{~d} x}$ if $y=\left(x^{2}+\frac{1}{x^{2}}\right)^{2}$Determine $\frac{\mathrm{d} y}{\mathrm{~d} x}$ if \ y=\left(x^{2}+\frac{1}{x^{2}}\right)^{2} \ ...
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Given $y=\sqrt{x^{7}}-\frac{5}{x^{3}}$ Which of the following is equivalent to $y$ ?
Given $y=\sqrt{x^{7}}-\frac{5}{x^{3}}$ Which of the following is equivalent to $y$ ?Given \ y=\sqrt{x^{7}}-\frac{5}{x^{3}} \ Which of the following is equivalent to $y$ ? \begin{align} \begin{array}{|l|l|} \hline \text { A } &amp ...
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Differentiate $g(t)=4(t+1)^{2}(t-3)$ with respect to $t$
Differentiate $g(t)=4(t+1)^{2}(t-3)$ with respect to $t$Differentiate &nbsp;$g(t)=4(t+1)^{2}(t-3)$ with respect to $t$ ...
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Differentiate the function $m=\frac{3}{2} n^{4}-1$
Differentiate the function $m=\frac{3}{2} n^{4}-1$Differentiate the function $m=\frac{3}{2} n^{4}-1$ ...
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Differentiate the function $y=3 x^{5}$
Differentiate the function $y=3 x^{5}$Differentiate the function $y=3 x^{5}$ ...
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$$\text { Consider the system } A X=B, \text { where } X=\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \text {. }$$
$$\text { Consider the system } A X=B, \text { where } X=\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \text {. }$$ \begin{aligned} &amp;\text { Consider the system } A X=B \text {, where } X=\left\begin{array}{l} x \\ y \\ z \end{array}\right \te ...
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The function $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ has
The function $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ has The function $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ has &nbsp; A. no extremum, B. exactly one maximum, C. exactly one minimum, D. exactly ...
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The function $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ is
The function $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ isThe function $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ is &nbsp; A. strictly increasing on $(-\infty, \infty)$, &nbsp; B. strictly increasing on $(-\infty ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 The function$f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$has ... 0 answers 3 views The function$f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$has ...The function$f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$has &nbsp; A. no point of inflection and is concave up on$(-\infty, \infty)$, &nbsp; B. no point o ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Prove Rolle's Theorem under the assumption that for the function$f$on$[a, b]$you have$f(x)>f(a)$for some$x$in the domain of$f$. 0 answers 5 views Prove Rolle's Theorem under the assumption that for the function$f$on$[a, b]$you have$f(x)>f(a)$for some$x$in the domain of$f$.Prove Rolle's Theorem under the assumption that for the function$f$on$a, b$you have$f(x)&gt;f(a)$for some$x$in the domain of$f$. ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Sketch the graph of$f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$0 answers 3 views Sketch the graph of$f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$Sketch the graph of$f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Evaluate the integral:$\int \dfrac{3 x+2}{\sqrt{4-3 x^{2}}} d x$0 answers 4 views Evaluate the integral:$\int \dfrac{3 x+2}{\sqrt{4-3 x^{2}}} d x$Evaluate the integral: &nbsp;&nbsp;$\int \dfrac{3 x+2}{\sqrt{4-3 x^{2}}} d x$... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Evaluate the integral:$\int_{0}^{\pi} 2 x \cos 2 x d x$0 answers 7 views Evaluate the integral:$\int_{0}^{\pi} 2 x \cos 2 x d x$Evaluate the integral:$\int_{0}^{\pi} 2 x \cos 2 x d x$... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Evaluate$\int \dfrac{\ln z}{(1+z)^{2}} d z$0 answers 5 views Evaluate$\int \dfrac{\ln z}{(1+z)^{2}} d z$Evaluate$\int \dfrac{\ln z}{(1+z)^{2}} d z$... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Evaluate the following integral:$\int \frac{t}{\sqrt{t^{4}-1}} d t$. 0 answers 4 views Evaluate the following integral:$\int \frac{t}{\sqrt{t^{4}-1}} d t$. Evaluate the following integral:$\int \dfrac{t}{\sqrt{t^{4}-1}} d t$. Hint:&nbsp;You may use the fact that$\int \sec \theta d \theta=\ln |\sec ...
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Evaluate the integral $\int_{1}^{2} \dfrac{2 s-2}{\sqrt{-s^{2}+2 s+3}} d s$
Evaluate the integral $\int_{1}^{2} \dfrac{2 s-2}{\sqrt{-s^{2}+2 s+3}} d s$Evaluate the integral $\int_{1}^{2} \dfrac{2 s-2}{\sqrt{-s^{2}+2 s+3}} d s$ ...
Evaluate $\int \dfrac{1}{\left(1+x^{2}\right) \arctan x} d x$
Evaluate $\int \dfrac{1}{\left(1+x^{2}\right) \arctan x} d x$Evaluate $\int \dfrac{1}{\left(1+x^{2}\right) \arctan x} d x$ ...