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Chapter Name :-
Matrices and Determinants

If [tex] A = \sf\left [\begin{array}{ccc}\omega&0\\0&\omega\end{array}\right][/tex] then find out value of [tex]\pink{\sf A ^{20226397596}}[/tex]


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Step-by-step explanation:

This is a inverse. matrix because we have alternating

zeroes and the w symbol it looks similar to the matrix,

[ 1 0]

[ 0 1]

Except, we just replace 1 with q. So we also know that if we multiply a inverse like matrix by itself, a infinite number times, it going to stay the same.

However, since we have variables in place for 1, we would raise the matrix to an nth power.

Because remeber that a inverse matrix has 1, an that

[tex]1 {}^{n} = 1[/tex]

So since we have a variable, we just raised it to the nth number.

We know that Zeroes raised to any power is 0, so w raised to the

^20226397596 is infact, w^20226397596, so our matrix is just

[w^20026397596 0]

[ 0 w^20026397596]

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The value for [tex]A^{20226397596}[/tex] is        [tex]\left[\begin{array}{cc}\omega^{20226397596}&0\\0&\omega^{20226397596}\end{array}\right] [/tex].

Procedure - Determination of the power of a matrix

Let be [tex]A[/tex] a diagonal matrix. By linear algebra we know that the n-th power of diagonal matrix of the form [tex]\left[\begin{array}{cc}\omega&0\\0&\omega\end{array}\right] [/tex] is equal to:

[tex]A^{n} = \left[\begin{array}{cc}\omega^{n}&0\\0&\omega^{n}\end{array}\right] [/tex], [tex]\forall\, n \ge 1[/tex], [tex]n\in \mathbb{N}[/tex] [tex]\forall \,\omega \in \mathbb{R}[/tex] (1)

Hence, we have the following result for [tex]A^{20226397596}[/tex]:

[tex]A^{20226397596} = \left[\begin{array}{cc}\omega^{20226397596}&0\\0&\omega^{20226397596}\end{array}\right] [/tex]

The value for [tex]A^{20226397596}[/tex] is        [tex]\left[\begin{array}{cc}\omega^{20226397596}&0\\0&\omega^{20226397596}\end{array}\right] [/tex]. [tex]\blacksquare[/tex]

To learn more on matrices, we kindly invite to check this verified question: https://brainly.com/question/4470545

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