Question

find all matrices that commute with the matrix A=[0 1]

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Adonis

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To find all matrices that commute with the matrix A, we need to find matrices B such that AB = BA.

Let B be the matrix:

B = \begin{bmatrix} a & b \ c & d \end{bmatrix}

Now, let's multiply AB and BA and set them equal to each other:

AB = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} c & d \ 0 & 0 \end{bmatrix}

BA = \begin{bmatrix} a & b \ c & d \end{bmatrix} \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & a \ 0 & c \end{bmatrix}

Setting AB and BA equal to each other:

\begin{bmatrix} c & d \ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & a \ 0 & c \end{bmatrix}

Comparing the entries, we get the following equations:

c = 0

d = a

This means that any matrix B that commutes with A has the form:

B = \begin{bmatrix} a & b \ 0 & a \end{bmatrix}

So, the matrices that commute with A are of the form:

\begin{bmatrix} a & b \ 0 & a \end{bmatrix}

where a and b can be any real numbers.

Answer: The matrices that commute with A are in the form\begin{bmatrix} a & b \ 0 & a \end{bmatrix} , where a and b can be any real numbers.

Let B be the matrix:

Now, let's multiply AB and BA and set them equal to each other:

Setting AB and BA equal to each other:

Comparing the entries, we get the following equations:

This means that any matrix B that commutes with A has the form:

So, the matrices that commute with A are of the form:

where a and b can be any real numbers.

Answer: The matrices that commute with A are in the form

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