Linear Algebra II Show that there do not exist square matric

Linear Algebra II

Show that there do not exist square matrices A, B such that AB - BA = I.

Solution

Dear student

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To prove

there do not exist square matrices A, B such that AB - BA = I.

Proof

Trace of a matrix is defined as sum of diagonal elements of a matrix

Now for LHS

Trace of AB-BA will always be 0

while for right hand side the trace f I shall never be 0

infact tract of I depends upon the rank of I.

Clearly LHS and RHS shall never be equal.

Hence proved