Where possible, find the product:

$\left[\begin{array}{cccc}2& 3& -1& 6\\ 0& 5& 4& 1\end{array}\right]\left[\begin{array}{cc}1& 3\\ 0& 2\end{array}\right]$

Cem Hayes
2020-12-30
Answered

Where possible, find the product:

$\left[\begin{array}{cccc}2& 3& -1& 6\\ 0& 5& 4& 1\end{array}\right]\left[\begin{array}{cc}1& 3\\ 0& 2\end{array}\right]$

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Find a basis for the space of $2\times 2$ diagonal matrices.

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

1. double column 1,

2. halve row 3,

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,

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.

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Is there a formula for the product of 3 matrices? That is, if $A\in {\mathbb{R}}^{m\times n},B\in {\mathbb{R}}^{n\times n},$, and $C\in {\mathbb{R}}^{n\times p}$, and I want the (i,j) entry of the product D=ABC, how can I write ${D}_{i,j}$? I know $(AB{)}_{i,j}=\sum _{k=1}^{n}{a}_{ik}{b}_{kj}$, but I'm not sure if this can be generalized to more than 2 matrices.

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7 = 1 mod 2

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Solve the system of equations using matrices. Use the Gaussian elimination method with back-substitution.

x+4y=0 x+5y+z=1 5x-y-z=79

x+4y=0 x+5y+z=1 5x-y-z=79

asked 2022-02-13

Let $A=\left[\begin{array}{ccc}1& 0& -2\\ -2& 1& 6\\ 3& -2& -5\end{array}\right]$ and $b=\left[\begin{array}{c}-1\\ 7\\ -3\end{array}\right]$ . Define a linear transformation T by T(x)=AX. Determine a vector x whose image under T is b. Is the vector x that you found unique or not? Explain your answer.