A=[ 2 3 -1 2 ] and f(x)=a x^2-b x+c ; a, b, c>0 are such that f(A)=0. Then (a)

A=[ 2 3 -1 2 ] and f(x)=a x^2-b x+c ; a,...

Key Concepts

Arithmetic Progression in Coefficients

An arithmetic progression is a sequence in which each term after the first is obtained by adding a fixed constant to the preceding term. When polynomial coefficients are described as being in arithmetic progression, it indicates a specific linear relationship among them. This concept is useful in problems where the structure of polynomial coefficients leads to symmetry or predictable patterns in the behavior of the polynomial.

Determinant and Its Relation to Eigenvalues

The determinant is a scalar attribute of a matrix that reflects important features such as invertibility and volume scaling in linear transformations. In the context of eigenvalues, the determinant of (A - xI) provides a polynomial whose zeros are the eigenvalues of A. This connection is vital for solving equations involving matrices and understanding their invariant properties.

Matrix Polynomial Evaluation

Evaluating a polynomial at a matrix involves substituting the matrix into the polynomial in place of the variable and using the identity matrix where a constant appears. For example, if f(x) = ax² - bx + c, then f(A) is defined as a·A² - b·A + c·I, where I is the identity matrix. This technique connects polynomial algebra with linear transformations and is useful for exploring matrix properties like eigenvalues or establishing annihilating polynomials.

Cayley-Hamilton Theorem

The Cayley-Hamilton theorem asserts that every square matrix satisfies its own characteristic polynomial. This result is central in linear algebra because it provides a natural polynomial relation that any matrix fulfills, thereby linking the matrix’s trace and determinant to its eigenvalues. It often serves as a tool to derive identities involving matrices or simplify computations related to matrix functions.

Characteristic Polynomial

The characteristic polynomial of a matrix is formed by taking the determinant of the matrix A minus x times the identity matrix, written as |A - xI|. This polynomial, typically expressed in terms of the trace and determinant for a 2x2 matrix, yields the eigenvalues of the matrix as its roots. It plays a key role in understanding the spectrum and dynamical behavior of linear transformations.

Transcript

00:01 So in the given question we have a matrix a that is given as 2 3 minus 1 2 right and we are given a function f of x and f of x is equal to a x square minus b x plus c and the coefficients a b br and the coefficients a br a, b and c are greater than 0, right? they are positive numbers and what else is given is that we are given that when we substitute the matrix a in this function we get f of a equal to 0.

00:51 So these are the details that we have been given in the question and we have a few options in the question as well.

01:00 The first option says that a, b, c.

01:03 Are in an arithmetic progression that is an ap.

01:09 In the second option it is told that the roots of the roots of f of x equal to 0 are 1 and 4 1 and 4.

01:30 Next we have the third option c in which we have a times the determinant of a minus x times i where i is the identity matrix which is of the same order of a and when we take this determinant what we get is f of x so this is what option c says and lastly we have option d in which it is said that the minimum value of determinant of a minus x i is 3a so these are the options that are given and we are told in the question that one or more of these options may be correct and we need to find which one of those options are which one of these options are correct and which ones are not right so first what we can do is we can take the function f of a equal to zero and substitute a in the function that is we are substituting a for x in the function f of x so a square a times small a times capital a square minus b a plus c i is equal to zero so we need to find a square first so a square is equal to we have the matrix a which is 2 3 minus 1 2 times 2 3 minus 1 2 when we take a square what we would get is you would get 2 times 2 4 minus 3 which is 1 we have 6 plus 6 which is 12 minus 2 minus 2 is minus 4 and minus 3 plus 4 is 1 so this is what we get as a square so this is what we get as a so let's substitute a square a and i in the above equation so we would have a times 1 12 minus 4 1 minus b times 2 3 minus 1 2 which is matrix a and we have c times i which is the identity matrix of which is the same order as that of a right so let's let's do this add, subtract these two matrices and let's do all these operations, right? so what we would have is we would have a matrix in which the elements are a minus 2b plus c, 12a minus 3d plus c times 0 is 0 and we have minus 4a minus 4a minus plus b plus c times 0 is again 0 and we have a minus 2b plus c in this position right so this is what we get as the matrix and we are told that this matrix is actually the zero matrix since f of a is equal to 0 so now what we can do is we can write a minus 2b plus c is then equal to 0 which means we can write a plus c is equal to 2 times b and from this what we can write is a plus c divided by 2 is equal to b so this result is a significant result because you see the first option in the question says that a b c are in ap right so if a b and c are in a, considering a as the first term, b as the second term and c as the third term, b would be let's say a plus d where d is the common difference and c would be a plus 2d and we can write a relation between a b and c such that a plus c divided by 2 is equal to a plus a plus 2 d divided by 2 which is equal to 2a plus 2d divided by 2.

06:29 So we can divide the numerator with 2 which would give us a plus c by 2 is equal to a plus d right and a plus d is in fact the number b.

06:42 So if a b c are in ap we can write this relation that a plus c by 2 is equal to b which means the first option is the is a correct option right so a b and c are in an ap so this is a correct option now next what we can check is let's take the second element in the first column of the matrix that we wrote so we would have minus 4 a plus b is equal to 0 from which we can write b is equal to 4a and similarly let's take the the relation a minus 2b plus c and substitute the value of b as 4a in this relation and we would have a minus 8b plus c equal to 0 from which we can write minus 7 it is 8a right yeah after substituting b equal to 4a we would have 2 times 4a which is 8a so minus 7a plus c equal to 0 or we can write c equal to 7a so we got b and c in terms of a so we can substitute these values in the equation a x square plus a x squared minus bx minus bx minus bx plus c and what we would have is we would have f of x is equal to a x square minus 4 a x plus 7a and from this we can take a as a as a common factor then it would be a times x squared minus 4x plus 7a and from this we can take a as a common factor then it would be a times x squared minus 4x plus 7 so this would be the simplified expression for f of x and now let's take the determinant a minus x i as we are told in the option in one of the options in the question and when we take this the matrix a is 2 3 minus 1 2 and x times identity matrix i 1 .01 would be x 0 0x and we have to take the determinant of this matrix right so this would be the determinant 2 minus x 3 minus 1 2 minus x right so this would be the determinant and when we take the determinant over here what we would get is 2 minus x squared minus minus 1 minus 1 times 3 which is minus 3 so minus of minus 3 we would get plus 3 right so we can expand this and we would get 4 minus 2 times 2 is 4 x plus x squared plus 3 and this is equal to x square minus 4x plus 7 right so in the option we can see that in the in the third option we have a times, a times we just got the result that a minus xi, the determinant of a minus xi is equal to x squared minus 4x plus 7 and when we multiply a on both sides we have a times a minus xi is equal to a times x squared minus 4x plus 7 which is in fact f of x right so in option c we can see that we have an answer that says a times a times determinant of a minus x i would give us f of x so this is what we proved right now so option c is also the correct option right next we have not checked option b right in option b we have f of x equal to 0 has roots 1 and 4 so when we take f of x equal to 0 we can take 8 times x squared minus 4x plus 7 equal to 0 and from this we can write x squared minus 4x plus 7 equal to 0 and to see whether 1 and 4 are the roots of this function we can just substitute 1 in this equation and it would be 1 minus 4 plus 7 which is equal to 7 minus 3 which is 4 and not equal to 0...

Please give Ace some feedback