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Integration of Piecewise Continuous Functions

Suppose A ≠...

Question

Suppose $A \neq 0, A^{2} = 0$ and $A (I + A)^{100} = k A$ , find $k$ .

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Solution

$G i v e n A 0, A^{2} = 0 a n d A (I + A)^{100} = k A {(I + A)}^{2} = (I + A) (I + A) = I + 2 A s i m i l a r l y {(I + A)}^{n} = I + n A A (I + A)^{100} = A (I + 100 A) = A I + 100 A^{2} = A H e n c e k = 1$

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a_{1}, a_{2}, . . ., a_{100}

be non-zero real numbers such that

a_{1} + a_{2} + . . . + a_{100} = 0

Q. Suppose A is a matrix such that

A^{2} = A

(I + A)^{10} = I + k A

k

.

A = [\begin{matrix} 1 & 3 3 & 4 \end{matrix}]

A^{2} - k A - 5 I_{2} = 0

k =

I = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

E = [\begin{matrix} 0 & 1 0 & 0 \end{matrix}]

(a I + b E)^{3} = a^{3} I + k a^{2} b E

Find the value of

k .

A = [\begin{matrix} 1 & 2 2 & 3 \end{matrix}]

A^{2} - K A - I_{2} = 0,

then value of k is-

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