5. If | 0 x-a x-b x+a 0 x-c x+b x+c 0 |=0, a ≠b ≠c, then

5. If | 0 x-a x-b x+a 0 x-c ...

Key Concepts

Factorization and Root Solving in Polynomial Equations

Once a determinant is expressed as a polynomial in a variable, factorization becomes a key tool for finding its roots. This step involves expressing the polynomial as a product of factors, allowing us to set each factor to zero independently and solve for the variable. It also helps in identifying different regimes or conditions under which the solutions are valid.

Determinant of a Matrix

A determinant is a scalar value that can be computed from the elements of a square matrix. It encodes important properties of the matrix, such as invertibility, and provides insights into systems of linear equations. In analysis, setting the determinant equal to zero often indicates points where the matrix is singular or where critical transitions occur in the system.

Expansion Methods for Determinants

Techniques such as cofactor expansion (Laplace expansion) are employed to calculate determinants, particularly for matrices of size 3×3 or larger. These methods allow the reduction of the determinant of a larger matrix into smaller, more manageable parts, which can then be simplified algebraically.

Parameterized Determinants and Algebraic Equations

When a matrix contains variable entries, its determinant becomes an algebraic expression (often a polynomial) in those variables. Setting the determinant equal to zero leads to equations whose solutions provide important insights, such as the values at which the matrix loses full rank or other critical phenomena occur.

Transcript

00:01 I have given a determinant this is equal to 0 a not equal to b not equal to c which is checked for the solution for x according to the given conditions so let's work on this if expanded the term line along r1 so we'll get here x negative a times we'll have x negative c x positive b negative we have x positive a then positive x negative b times x positive a x positive c that equals 0 this we just work on this so let's work on this here it is 0 x positive way multiply with 0 we have so we'll have this so can you factor out anything x negative b next positive c now we cannot factor out just have to expand it so let expand this we'll have x negative a times x squared plus bx negative cx negative bc plus x negative bc plus x square plus x plus a x plus a c that equals zero spend this further so we'll get x cube x cube plus b x square negative c x square negative bc x square negative ab x positive a x x positive abc and here we have positive x cube cx x square a x square a cx square ac negative b x square negative bc x negative bc x negative ab x and negative a bcc this cancellations here this will be negative 2 ab x negative bc x cancellations cancellations so we have to be 2x cube we have 2x cube and then negative 2 bcx negative 2ab x positive 2 ac x that equals 0 in fact of 2 x we get x square plus ac c negative ab negative bc that equals 0 so from here we get the factors it's coming out to be the factors are here we have x equals 0 where we have x equals b a plus c negative ac this we have now if b a plus c negative ac this is greater than 0 if this part is greater than 0 then we will have two real roots from here and two real roots from here this correspond to three real roots it will give three roots and if we have b a plus c negative a .c is less than 0.

03:40 This will give imaginary roots in that case and this will give one real root.

03:44 So, one root in that case.

03:46 The imaginary roots are not counting...

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