Problem 45 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(A\) satisfies the equation \(x^{3}-5 x^{2}+4 x+k I=0\), then \(A^{-1}\) exists if (A) \(k \neq-1\) (B) \(k \neq 0\) (C) \(k \neq 1\) (D) None of these

Step-by-Step Solution

Answer

The matrix \(A\) is invertible if \(k \neq 0\).

Step 1: Understanding the Problem

We need to determine under what condition the matrix \(A\) is invertible. For a matrix \(A\) to have an inverse (i.e., \(A^{-1}\) exists), its determinant must not be zero. We were given a matrix polynomial equation involving \(A\). Let's analyze this polynomial to find when the determinant of \(A\) is non-zero in terms of \(k\).

Step 2: Analyzing the Given Equation

The given equation is \(x^{3} - 5x^{2} + 4x + kI = 0\). Think of this in the form of a characteristic polynomial \( \ ext{det}(xI - A) = 0\) where \(I\) is the identity matrix and \(x\) is a scalar.

Step 3: Characterizing Matrix Invertibility

A matrix \(A\) is invertible if its determinant \(\det(A) eq 0\). For \(A\) to be invertible, none of the roots of its characteristic equation should be 0. This means \(k\) should be chosen such that the polynomial doesn't have zero as a root.

Step 4: Applying the Invertibility Condition

To find \(k\) such that \(0\) is not a root of the characteristic polynomial \(x^3 - 5x^2 + 4x + k = 0\), substitute \(x = 0\) into the polynomial. This gives: \(k = 0^{3} - 5 \cdot 0^{2} + 4 \cdot 0 + k = k\). So, for the polynomial to have \(0\) as a root, \(k\) must be \(0\).

Step 5: Determining the Condition for \(A\) to Have an Inverse

\(A\) will not be invertible if \(k = 0\), as that would make \(0\) a root (causing the determinant to be zero). Therefore, for \(A\) to be invertible, we need \(k eq 0\). This corresponds to option (B).

Key Concepts

Characteristic Polynomial

The characteristic polynomial is a key concept in linear algebra associated with square matrices. It is a special polynomial derived from a given matrix and is instrumental in understanding the properties of the matrix, such as eigenvalues, which are roots of this polynomial. The characteristic polynomial for a matrix \(A\) of size \(n\times n\) is given by the determinant \(\det(\lambda I - A)\), where \(I\) is the identity matrix of the same size, and \(\lambda\) is a scalar.
The roots of this polynomial reveal crucial insights about the matrix. In the context of the problem, we use the given polynomial \(x^3 - 5x^2 + 4x + k = 0\). This polynomial can be imagined as a characteristic polynomial, which we should analyze to find conditions for invertibility. The roots indicate where the determinant might be zero, affecting the matrix's invertibility.

Determinant of Matrix

The determinant of a matrix is a fundamental scalar value that can tell us a lot about the matrix's properties. For instance, it is a key factor in determining whether a matrix is invertible. Specifically, a matrix \(A\) is invertible if and only if its determinant \(\det(A)eq0\). Should the determinant equal zero, the matrix is considered singular, meaning it lacks an inverse.
In the given problem, determining the condition where the determinant of the matrix is zero involves analyzing the polynomial \(x^3 - 5x^2 + 4x + k = 0\). We substitute values of \(x\) (in particular, \(x = 0\)) to explore the relationship between \(k\) and the determinant, as the determinant of a matrix is inevitably tied to its characteristic equation.

Roots of Polynomial

The roots of the polynomial are the solutions to the equation \(x^3 - 5x^2 + 4x + k = 0\). These roots play a critical role in determining the properties of the matrix. Should any of these roots be zero, it indicates that the determinant of the matrix equals zero, thereby implying the matrix is not invertible.

• When solving for roots, set \(x = 0\) to calculate if zero is a root.
• If zero is a root, it implies \(k = 0\).

In essence, a matrix can only be invertible if zero is not one of its roots since a root of zero would compromise its determinant's value, leading to non-invertibility.

Invertible Matrix Condition

The invertibility condition of a matrix directly relates to its determinant. That is, a matrix is invertible if its determinant is not zero. This criterion is closely connected to the characteristic polynomial, as the determinant's zero values correspond to roots of the polynomial.
In the exercise, we pose the question of when matrix \(A\) is invertible. We identified that \(k\) should not equal zero since zero as a root would result in a determinant of zero. Therefore, the condition for matrix \(A\) to have an inverse is succinctly given by \(k eq 0\). Recognizing this leads to the understanding that among the options provided, option (B) is correct, maintaining the invertibility of matrix \(A\).