If | a x-b y-c z a y+b x c x+a z a y+b x b y-c z-a x b z+c y c x

If | a x-b y-c z a y+b x c x+a z a y+b x ...

Key Concepts

Vector Dot Product and Quadratic Forms

The dot product is an operation that takes two vectors and returns a single number, representing the product of their magnitudes and the cosine of the angle between them, which is key in expressing projections and orthogonality. Quadratic forms, like the sum of squares of vector components, represent the squared magnitude (or norm) of a vector. When these appear in determinant identities, they can often be factored out, revealing deeper connections between the algebraic structure and geometric interpretations.

Matrix Factorization Techniques

Matrix factorization techniques involve expressing a complicated determinant as a product of simpler, more manageable factors by exploiting structural symmetries, linear dependencies, or known algebraic identities. This approach is particularly useful in simplifying and evaluating determinants that contain both linear and quadratic terms.

Determinant Properties

A determinant is a scalar value computed from a square matrix that encapsulates important geometric and algebraic information about the matrix such as scaling factors, invertibility, and volume change under the linear transformation described by the matrix. Its properties such as multilinearity and the alternating sign with respect to row swaps make it a powerful tool in analyzing systems of linear equations and transformations.

Transcript

00:01 So in the given question we are given that the matrix given as the determinant of the matrix given as a x minus b y minus z a y plus b x x x plus a x x plus a y plus b x b y plus b x b y minus c z minus a x b z plus c y and we have the third row c x plus a z bz plus c y cz plus c z z minus a x minus b y the determinant of this matrix is equal to k times a square plus b square plus c square times a x plus b y plus c z if this is given in the question we are told to find what is the value of k so first let's take the given matrix that is given in the question the determinant that is given in the question and we should simplify this right so what we are going to do first is we can take a as a common factor from the first row of the given determinant right or instead of taking a common factor just multiply and divide the first row with a right so we can keep the a that we are going to divide it with outside the determinant and we are going to multiply the first row with a.

02:23 So what we would have is a square x minus a b y minus a c z.

02:38 Second row we have a square y plus a b x and in the third row we have a cx plus a square so this is what we do first right we divide and multiply with with the first row the first column of the determinant so only the first column is changing the rest of the elements are the same so let's write them again so this is a y plus bx bx b minus z x minus a x b z plus c y cx plus a z b z plus c y and cz minus a x minus b z plus c y and cz minus a x minus b y in the third row third element in the next step we are doing a column operation which is c1 changes to c1 plus bx c2 plus cx c3.

04:08 And once we do this what we get as the new matrix is 1 by a and the first column we would have a square plus b squared plus c squared times x in the second row first column we would have the same a square plus b square plus c squared times y and over here also we have a square plus b square plus c squared times z the rest of the determinant is the same right so this is what we have c a c x plus a z b z plus c y plus c y plus c y z minus a x minus by.

05:10 Now in the next step what we have to do is to take a square plus b square plus c square from the first row as a common factor.

05:20 So we would have a square plus b square plus c square divided by a and in the determinant we would have x, y, z and this part of the matrix is unchecked.

05:38 Changed right this is the same part again so you could just copy this part and place it over here so let's place the determinant over here right so we have the first row it is all the columns are same except the first column which has a x y z now in the next step what we do is we can take we can multiply and divide the first row with x right so if we do that what we would have is let's take this let's copy this again so we can copy this and we can paste it over here and the next step what we do is is we take, we divide and multiply the determinant with x and we multiply the first row of the determinant right.

07:04 So x square and we have a y x or a xy plus v x square plus c and c and c x square plus a z x and in the next step what we can do is to take a row operation a row operation that says r1 changes to r1 changes to r1 plus y times x times r2 plus z times x times are 3.

07:51 So when we do this what we get is we have a square plus b squared plus c squared divided by a x over here and inside the determinant we have x squared plus y square plus z squared x squared plus y square plus z square times b and x squared plus x squared plus y square plus z square times c so the aim of this method is to simplify the determinant as far as possible as much as possible so that we can take common factors outside the determinant and solve this determinant right so if we solve it right now it would be so hard to expand this determinant so we would do some row of and column operations to simplify the matrix, take some common factors out and then when the determinant is simplified we would take, we can take, we can calculate the determinant easily.

09:13 So that is the method that we are going to use...

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