# Recent questions tagged determinant

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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 0 answers 2 views 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 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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If $A$ and $B$ can be simultaneously diagonalized, show that $A B=B A$.
If $A$ and $B$ can be simultaneously diagonalized, show that $A B=B A$.Two matrices $A, B$ can be simultaneously diagonalized if there is an invertible matrix that diagonalizes both of them. In other words, if there is ...
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Give an example of a matrix $A$ with the following three properties:
Give an example of a matrix $A$ with the following three properties:Give an example of a matrix $A$ with the following three properties: i). $A$ has eigenvalues $-1$ and 2 . ii). The eigenvalue $-1$ has eigenve ...
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Compute the determinant of $A^{10}+A$.
Compute the determinant of $A^{10}+A$.a) Find a $2 \times 2$ real matrix $A$ that has an eigenvalue $\lambda_{1}=1$ with eigenvector $E_{1}=$ $\left(\begin{array}{l}1 \\ 2\end{arr ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 [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 $A(t)$ be a family of invertible real matrices depending on the real parameter $t$ and assume they are invertible.
Let $A(t)$ be a family of invertible real matrices depending on the real parameter $t$ and assume they are invertible.Let $A(t)$ be a family of invertible real matrices depending on the real parameter $t$ and assume they are invertible. Show that the inverse matri ...
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Let $A$ and $B$ be $n \times n$ matrices. If $A+B$ is invertible, show that $A(A+B)^{-1} B=$ $B(A+B)^{-1} A$. [Don't assume that $A B=B A$ ].
Let $A$ and $B$ be $n \times n$ matrices. If $A+B$ is invertible, show that $A(A+B)^{-1} B=$ $B(A+B)^{-1} A$. [Don't assume that $A B=B A$ ].Let $A$ and $B$ be $n \times n$ matrices. If $A+B$ is invertible, show that $A(A+B)^{-1} B=$ $B(A+B)^{-1} A$. Don't assume that $A B=B A ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Think of the matrix \(A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ as mapping one plane to another.
Think of the matrix $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ as mapping one plane to another.Think of the matrix $A=\left(\begin{array}{ll}a &amp; b \\ c &amp; d\end{array}\right)$ as mapping one plane to another. a) If two lines in the firs ...
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Prove that $A$ is invertible and $B A=I$ as well.
Prove that $A$ is invertible and $B A=I$ as well.Suppose that $A$ is an $n \times n$ matrix and there exists a matrix $B$ so that \ A B=I \text {. } \ Prove that $A$ is invertible and $B 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, 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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Let $A=\left(\begin{array}{rrr}1 & 1 & -1 \\ 1 & -1 & 2\end{array}\right)$
Let $A=\left(\begin{array}{rrr}1 & 1 & -1 \\ 1 & -1 & 2\end{array}\right)$Let $A=\left(\begin{array}{rrr}1 &amp; 1 &amp; -1 \\ 1 &amp; -1 &amp; 2\end{array}\right)$ a) Find the general solution $\mathbf{Z}$ of the homog ...
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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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Let $A$ and $B$ be $n \times n$ matrices. Which of the following is true?
Let $A$ and $B$ be $n \times n$ matrices. Which of the following is true?Let $A$ and $B$ be $n \times n$ matrices. Which of the following is true? &nbsp; A) $\operatorname{det}(A)=\operatorname{det}(-A)$ B) $\operatornam ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Which of the following operations does not change the determinant? 0 answers 4 views Which of the following operations does not change the determinant?Which of the following operations does not change the determinant? &nbsp; A) interchanging two rows B) multiplying a row by a constant C) adding t ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Find the inverse of the matrix $$A=\left(\begin{array}{lll} 4 & 3 & 4 \\ 5 & 4 & 6 \\ 4 & 3 & 3 \end{array}\right)$$ 0 answers 3 views Find the inverse of the matrix $$A=\left(\begin{array}{lll} 4 & 3 & 4 \\ 5 & 4 & 6 \\ 4 & 3 & 3 \end{array}\right)$$Find the inverse of the matrix $$A=\left(\begin{array}{lll} 4 &amp; 3 &amp; 4 \\ 5 &amp; 4 &amp; 6 \\ 4 &amp; 3 &amp; 3 \end{array}\right)$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Calculate the determinant $$\left|\begin{array}{rrrr} 4 & 2 & 3 & 4 \\ 1 & 0 & 1 & 2 \\ 1 & -1 & 1 & 0 \\ 5 & 1 & 4 & 3 \end{array}\right|$$ 0 answers 8 views Calculate the determinant $$\left|\begin{array}{rrrr} 4 & 2 & 3 & 4 \\ 1 & 0 & 1 & 2 \\ 1 & -1 & 1 & 0 \\ 5 & 1 & 4 & 3 \end{array}\right|$$Calculate the determinant$$\left|\begin{array}{rrrr} 4 &amp; 2 &amp; 3 &amp; 4 \\ 1 &amp; 0 &amp; 1 &amp; 2 \\ 1 &amp; -1 &amp; 1 &amp; 0 \\ 5 &amp; ... close 0 answers 4 views Which of the following systems has augmented matrix $$\left[\begin{array}{cccc} 1 & 4 & 0 & 6 \\ 4 & 2 & -1 & 0 \\ 0 & 1 & 5 & -6 \end{array}\right] \text { ? }$$Which of the following systems has augmented matrix$\left\begin{array}{cccc}1 &amp; 4 &amp; 0 &amp; 6 \\ 4 &amp; 2 &amp; -1 &amp; 0 \\ 0 &amp; 1 &am ...
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$$\text { The rank of the coefficient matrix } A=\left[\begin{array}{ccc} 1 & 2 & -3 \\ 3 & -1 & 5 \\ 4 & 1 & 2 \end{array}\right] \text { equals: }$$ \text { The rank of the coefficient matrix } A=\left\begin{array}{ccc} 1 &amp; 2 &amp; -3 \\ 3 &amp; -1 &amp; 5 \\ 4 &amp; 1 &amp; 2 ...
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The augmented matrix of a system $A X=B$ has been transformed using elementary row operations to ...
The augmented matrix of a system $A X=B$ has been transformed using elementary row operations to ... \begin{aligned} &amp;\text { The augmented matrix of a system } A X=B \text { has been transformed using elementary row }\\ &amp ...
$$\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 ...