y=sina x and yr denote rth derivative of y . If A=[ y y1 y2 y1 y2 y3

Y=sina x and yr denote rth derivative of y . If A=[ y ...

Key Concepts

Matrix Invertibility and Singularity

A matrix is invertible (non-singular) if and only if its determinant is nonzero. Issues of singularity arise when the matrix's rows or columns become linearly dependent. In the given context, understanding when the matrix loses its invertibility, based on parameters or evaluation points, is key to addressing the problem's assertions about the existence of an inverse.

Properties of Symmetric Matrices

A symmetric matrix is one that is equal to its transpose, meaning its (i, j)th entry is the same as its (j, i)th entry for all i and j. This property influences many aspects of a matrix's behavior including eigenvalues and stability, and is relevant here since the matrix formed by the sinusoidal function and its derivatives is constructed to be symmetric.

Cyclic Nature of Trigonometric Function Derivatives

The derivatives of sine and cosine functions follow a repeating cycle every four derivatives. For example, differentiating sin(ax) four times brings one back to sin(ax) up to a constant multiple. Recognizing this cycle is important for understanding the structure of the higher order derivatives used in the matrix.

Derivatives of Sinusoidal Functions

This concept involves finding successive derivatives of sinusoidal functions such as sin(ax). When differentiating sin(ax), the result includes cosine functions and factors of a, with signs alternating and powers of a increasing with each derivative. This is fundamental for forming the matrix whose entries are these derivatives.

Transcript

00:01 So in the given question we are told that y is equal to y equal to sign of a x right and we are told that y r is the earth derivative of y yr denotes the earth derivative of y and if that's the case we are given a matrix a in which the elements are y, y, y, 1, y2, y2, y2, y3, y2, y2, y3, y4.

00:54 So this is the matrix that is given and we have a few statements that are given in the question.

01:01 So the options given in the question are that if these are given then the matrix a is a non -singular matrix is non -singular.

01:21 Non -singular means the determinant of a is not equal to 0 when a equal to 1.

01:29 This is one of the options.

01:31 Option b is b is a says that a is a symmetric matrix.

01:38 A is symmetric for any value of x.

01:51 And the third option c says that the inverse of a does not exist, does not exist when a times x x, when a times x equal to pi by 2 and we have the last option which is d which says a inverse does not exist does not exist when a times x is equal to pi so we should choose we should choose the correct answer we should find the correct answer out of these four options that are given in the question right so what we can do over here is to first take the find the the first second third and fourth derivative of sign of a x right so we know y is equal to sign of a x then y 1 that is the first derivative of sine of a x would be a times a times cost of a x the second derivative would be minus a square sign of a x the third derivative would be minus a cube cost of a x and the fourth derivative would be a fourth derivative would be a to the power 4 sine of a x so we can substitute these values in the matrix a and we would have sine a x a cos a x and by 2 is minus a square sine of a x in in the second row, in the second row we have, y1 which is a cost of a x minus a square sine of a x and y3 which is minus a cube cost of a x.

04:40 Next we have y2 in the third row which is minus a square sine of a x.

04:48 Then we have y3 which is minus a cube cost of a x minus a cube cost of a x and at last we have a to the power 4 sign a x as y 4 now what we can do is we need to simplify this matrix right so we can take the value we can we can see that in this matrix when we take the transpose of this matrix, we would get the matrix a itself, right? so when we take the transpose of this matrix, we get a itself, which means a is a symmetric matrix, right? a is symmetric and since it is a symmetric matrix, we can write now that we have found that the a is symmetric for any value of a for any value of x for any value of x the matrix a would be symmetric since when we took a transpose we got a itself as the matrix right so now what we can do is we can just substitute the value of a as one right so when we put a equal to 1 in the matrix what we would get is we would have a equals sine x cos x sine x x x minus x minus x minus x minus x minus x and we have minus sine x minus x and we have minus sine x minus cox and and sine of x...

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