Calculate ${B}^{10}$ when

$$B=\left(\begin{array}{cc}1& -1\\ 1& 1\end{array}\right)$$

$$B=\left(\begin{array}{cc}1& -1\\ 1& 1\end{array}\right)$$

imchasou
2022-09-19
Answered

Calculate ${B}^{10}$ when

$$B=\left(\begin{array}{cc}1& -1\\ 1& 1\end{array}\right)$$

$$B=\left(\begin{array}{cc}1& -1\\ 1& 1\end{array}\right)$$

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Let B be a 4x4 matrix to which we apply the following operations:

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3. add row 3 to row 1,

4. interchange columns 1 and 4,

5. subtract row 2 from each of the other rows,

6. replace column 4 by column 3,

7. delete column 1 (column dimension is reduced by 1).

(a) Write the result as a product of eight matrices.

(b) Write it again as a product of ABC (same B) of three matrices.

1. double column 1,

2. halve row 3,

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4. interchange columns 1 and 4,

5. subtract row 2 from each of the other rows,

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7. delete column 1 (column dimension is reduced by 1).

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(b) Write it again as a product of ABC (same B) of three matrices.

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Write the given matrix equation as a system of linear equations without matrices. $\left[\begin{array}{cc}3& 0\\ -3& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}6\\ -7\end{array}\right]$

asked 2021-02-16

If $3\left[\begin{array}{cc}{x}_{1}& {x}_{2}\\ {x}_{3}& {x}_{4}\end{array}\right]=\left[\begin{array}{cc}{x}_{1}& 2\\ -1& 4{x}_{4}\end{array}\right]+\left[\begin{array}{cc}4& {x}_{1}+{x}_{2}\\ {x}_{3}+{x}_{4}& 3\end{array}\right]$

1.${x}_{1}=-2,{x}_{2}=2,{x}_{3}=-2,{x}_{4}=-3$

2.${x}_{1}=2,{x}_{2}=-2,{x}_{3}=-2,{x}_{4}=-3$

3.${x}_{1}=2,{x}_{2}=2,{x}_{3}=2,{x}_{4}=-3$

4.${x}_{1}=2,{x}_{2}=2,{x}_{3}=-2,{x}_{4}=3$

5,${x}_{1}=2,{x}_{2}=2,{x}_{3}=-2,{x}_{4}=-3$

1.

2.

3.

4.

5,

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Find all 2 by 2 matrices that are orthogonal and also symmetric. Which two numbers can be eigenvalues of those two matrices?

asked 2022-09-09

Use the relation $|AB|=|A||B|$ to show that

$$({a}_{1}^{2}+{a}_{2}^{2})({b}_{1}^{2}+{b}_{2}^{2})=({a}_{1}{b}_{1}-{a}_{2}{b}_{2}{)}^{2}+({a}_{2}{b}_{1}+{a}_{1}{b}_{2}{)}^{2}.$$

$$({a}_{1}^{2}+{a}_{2}^{2})({b}_{1}^{2}+{b}_{2}^{2})=({a}_{1}{b}_{1}-{a}_{2}{b}_{2}{)}^{2}+({a}_{2}{b}_{1}+{a}_{1}{b}_{2}{)}^{2}.$$

asked 2021-02-27

Find if possible the matrices:

a. AB b. BA.

$A=\left[\begin{array}{ccc}1& -1& 4\\ 4& -1& 3\\ 2& 0& -2\end{array}\right],B=\left[\begin{array}{ccc}1& 1& 0\\ 1& 2& 4\\ 1& -1& 3\end{array}\right]$

a. AB b. BA.