If 2 s=a+b+c and | a^2 (s-a)^2 (s-a)^2 (s-b)^2 b^2 (s-b)^2 (s-c)^2

If 2 s=a+b+c and | a^2 (s-a)^2 (s-a)^2 ...

Key Concepts

Symmetric Algebraic Expressions

Many determinants, especially those with structured or patterned elements, exhibit symmetry. Recognizing the symmetry in algebraic expressions can simplify the evaluation process, often allowing one to factor the determinant into a product of similar terms. This approach is useful in reducing complex expressions into more manageable, factorized forms.

Polynomial Factorization

Factorization in algebra involves expressing a polynomial as a product of simpler polynomials or factors. Identifying common factors in a determinant that is expanded into a polynomial form is crucial in simplifying the expression and relating it to a product form, as one does in comparing two algebraic expressions.

Determinants

The concept of determinants is central in linear algebra and is used to compute a scalar value from a square matrix. Determinants have properties such as linearity, the effect of row operations, and multiplicative behavior under matrix multiplication, which make them useful in solving systems of equations and analyzing matrix invertibility.

Matrix Row Operations and Properties

Evaluating determinants often involves applying row or column operations to simplify the matrix. This includes techniques like adding multiples of one row to another or factorizing common elements from rows or columns. These operations take advantage of the multilinearity of the determinant function and help to reveal underlying patterns in the matrix.

Transcript

00:01 I have given that 2 s equals a plus b plus t we're given here the determinant that equals k times s negative a as negative b as negative t we just find the value of k here so let's go with that what we can do we can assume in this we'll take let's say s negative a equals alpha this we take now so next we take as negative b equals beta next we take s negative c equals gum then we'll have beta plus gamma, there'll be 2s negative b plus t or 2s negative.

00:49 Now b plus 3 that is given as 2 s negative a, there will be a.

00:54 So beta plus gamma equals a.

00:57 Similarly, we'll have gamma plus alpha that equals b, alpha plus beta, that equals b.

01:04 Alpha plus beta, that equal t.

01:05 Also we have alpha plus beta plus gamma that equals 3s negative a plus a plus a plus.

01:12 B plus b plus c that is giving three s negative 2s that equals that is clear now the determinant is the coming after b we put the value of a as negative a as negative b as negative c so determinant will become left -hand side that equals we get beta plus gamma 4 square alpha square alpha square then beta square gamma plus alpha square then beta square gamma plus alpha 4 square then beta square if we have gamma square gamma square then alpha plus beta 4 square is we got here so if you just work on this we'll apply here in this determinant we will apply c1 tends to c1 negative c2 and c2 102 2 2 negative c3 so apply that so, we get to be beta plus gamma ho square, negative alpha square, then beta square negative gamma plus alpha hour square.

02:30 This would be 0.

02:32 Next is coming out to be this will give 0.

02:35 Then gamma plus alpha square negative beta square.

02:40 That will give gamma square negative alpha plus beta whole square.

02:46 Here we have alpha square, beta square, alpha plus beta square.

02:54 Which we have got here.

02:56 Now, from here we get, we'll get beta plus gamma plus alpha times beta plus gamma negative alpha, beta plus gamma plus alpha, beta negative gamma negative alpha.

03:17 So we get 0...

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