Verify the formula in (1) for an arbitrary positive integer n by induction. [HINT: Expand in min

Verify the formula in (1) for an arbitrary positive integer...

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Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to prove statements or formulas that are asserted for all positive integers. It involves two main steps: verifying the base case, usually for n = 1, and proving that if the statement holds for an arbitrary integer n, then it also holds for n + 1. This method establishes the truth of the statement for every positive integer.

Cofactor Expansion

Cofactor expansion is a method for calculating the determinant of a matrix by expanding it along a specific row or column. This involves multiplying each element of the chosen row or column by its corresponding cofactor—the determinant of the submatrix that remains after removing the element’s row and column, along with an alternating sign—and summing these products.

Minors

Minors are the determinants of the submatrices obtained by deleting one or more rows and columns from a larger square matrix. In the context of cofactor expansion, each minor is used to compute a cofactor, which in turn contributes to the overall calculation of the full determinant.

Transcript

00:01 Okay, i want to prove this by induction and i want n to take values which are natural numbers.

00:09 Or the first natural number is 1.

00:12 So we'll test for n equals 1 and see if it works.

00:18 So test for n equals 1.

00:22 That's the first step.

00:24 The left hand side, if n is 1, then we simply have 5 negative 8, 2.

00:35 Minus 3 as the matrix power 1 could be here but not required.

00:45 In the right hand side replace the n by 1 i will get 4 plus 1 is the first top left then negative 8 and 2n is clearly 2 and then 1 minus 4 n is 1 minus 4 and that is the same thing you get 5 negative 8, 2 negative 3 and that equals the left -hand side.

01:30 Hence expression is true for n equals 1.

01:45 Okay next test for n equals k.

01:57 Sorry, that should be assumed true for n equals k.

02:01 So we'll change that and put assume true for n equals k.

02:15 Other words this we assume is true.

02:20 Just replace the n here by k on both sides.

02:28 Then if this is true then so is the following.

02:44 The same thing as before and we post multiply on both sides by 5 minus 8 to minus 3.

02:55 In other words a valid algebraic operation on both sides.

03:00 Same thing on both sides.

03:02 So if this is true, so is all of this.

03:08 The keyword here is of course if...