यदि `A=[{:(1,1,1),(1,1,1),(1,1,1):}]`, तो सिद्ध कीजिए कि `A^(n)=[{:(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1)):}],ninN`.

Explanation

यहाँ `A^(n)=[{:(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1)):}]` n = 1 के लिये ,
`A=[{:(3^(0),3^(0),3^(0)),(3^(0),3^(0),3^(0)),(3^(0),3^(0),3^(0)):}]=[{:(1,1,1),(1,1,1),(1,1,1):}]` जो सत्य है ।
`thereforeA^(n),n=1` के लिये सत्य है ।
माना `A^(n),n=k` के लिये सत्य है ।
`thereforeA^(k)=[{:(3^(k-1),3^(k-1),3^(k-1)),(3^(k-1),3^(k-1),3^(k-1)),(3^(k-1),3^(k-1),3^(-1)):}]`
n = K +1 के लिये, `A^(k+1)=A^(k)*A`
`=[{:(3^(k-1),3^(k-1),3^(k-1)),(3^(k-1),3^(k-1),3^(k-1)),(3^(k-1),3^(k-1),3^(k-1)):}][{:(1,1,1),(1,1,1),(1,1,1):}]`
`=[{:(3^(k-1)+3^(k-1)+3^(k-1),3^(k-1)+3^(k-1)+3^(k-1),3^(k-1)+3^(k-1)+3^(k-1)),(3^(k-1)+3^(k-1)+3^(k-1),3^(k-1)+3^(k-1)+3^(k-1),3^(k-1)+3^(k-1)+3^(k-1)),(3^(k-1)+3^(k-1)+3^(k-1),3^(k-1)+3^(k-1)+3^(k-1),3^(k-1)+3^(k-1)+3^(k-1)):}]`
`=[{:(3*3^(k-1),3*3^(k-1),3*3^(k-1)),(3*3^(k-1),3*3^(k-1),3*3^(k-1)),(3*3^(k-1),3*3^(k-1),3*3^(k-1)):}]=[{:(3^(k),3^(k),3^(k)),(3^(k),3^(k),3^(k)),(3^(k),3^(k),3^(k)):}]`
`therefore^(n),n = k+1` के लिये भी सत्य है ।
अतः `A^(n),n` के सभी प्राकृतिक मानों के लिये सत्य है । यही सिद्ध करना था ।