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Question

If A = diag (a, b, c), show that An = diag (an, bn, cn) for all positive integer n.

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Solution

We shall prove the result by the principle of mathematical induction on n.

Step 1: If n = 1, by definition of integral power of a matrix, we have

A1=a1000b1000c1=a000b000c=A

So, the result is true for n = 1.

Step 2: Let the result be true for n = m. Then,

Am=am000bm000cm ...(1)

Now, we shall check if the result is true for n=m+1.
Here,
Am+1=am+1000bm+1000cm+1

By definition of integral power of matrix, we have

Am+1=AmAAm+1=am000bm000cma000b000c From eq. 1Am+1=aam+0+00+0+00+0+00+0+00+bbm+00+0+00+0+00+0+00+0+ccmAm+1=am+1000bm+1000cm+1

This shows that when the result is true for n = m, it is also true for n=m+1.
Hence, by the principle of mathematical induction, the result is valid for any positive integer n.

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