Problem 56 - Algebra for JEE ADVANCED

Question

If \(A\) is non-singular and \((A-2 I)(A-4 I)=O\), then \(\frac{1}{6} A+\frac{4}{3} A^{-1}\) is equal to a. \(O\) b. \(I\) c. \(2 I\) d. \(6 I\)

Step-by-Step Solution

Answer

The expression evaluates to \(I\), option b.

Step 1: Understand the Null Product

We start with \((A - 2I)(A - 4I) = O\).This implies that the matrix \(A\) satisfies the characteristic equation with roots 2 and 4, making it a matrix equivalent to \(2I\) and \(4I\).

Step 2: Eigenvalue Analysis

Since the product is zero, it means \(A = 2I\) or \(A = 4I\), depending on the eigenvalues. As \(A\) is non-singular, it cannot be zero. Therefore, \(A\) is likely a matrix made up of its eigenvalues arranged diagonally along the main diagonal line if it's diagonalizable.

Step 3: Use Possible Eigenvalues

Given \((A - 2I)(A - 4I) = O\), there are two possibilities: eigenvalues of 2 or 4. It suggests that for these eigenvalues, either \(A = 2I\) or \(A = 4I\) ensures \((A-2I)\) or \((A-4I)\) contributes to the null product.

Step 4: Calculate the Desired Expression

We need to compute \(\frac{1}{6}A + \frac{4}{3}A^{-1}\).If we let \(A = 2I\) first, then\[ A^{-1} = (2I)^{-1} = \frac{1}{2}I \]Substituting into \[\frac{1}{6}(2I) + \frac{4}{3} \left(\frac{1}{2}I\right) = \frac{1}{3}I + \frac{2}{3}I = I\]

Step 5: Check Consistency with Second Eigenvalue

As an additional check, assume \(A = 4I\), hence \[ A^{-1} = (4I)^{-1} = \frac{1}{4}I \]and replace\[ \frac{1}{6}(4I) + \frac{4}{3}(\frac{1}{4}I) = \frac{2}{3}I + \frac{1}{3}I = I \]Both scenarios check out and yield \(I\).

Step 6: Solution Selection

Both cases result in the expression resolving to \(I\), confirming that the expression, given the non-singular and eigenvalue conditions, resolves to the identity matrix \(I\).

Key Concepts

Eigenvalues

In the context of matrix algebra, eigenvalues are special numbers associated with a square matrix. They play a crucial role in understanding the properties of the matrix. Given a matrix \( A \), an eigenvalue of \( A \) is a scalar \( \lambda \) such that there exists a nonzero vector \( \mathbf{v} \) satisfying:

• \( A\mathbf{v} = \lambda\mathbf{v} \)

The vector \( \mathbf{v} \) is known as an eigenvector, and the equation above shows that applying the matrix to an eigenvector results in a scalar multiple of that vector itself.
In the provided exercise, the matrix \( A \) has eigenvalues 2 and 4. This is derived from the equation \((A - 2I)(A - 4I) = O\), where \( O \) is the zero matrix. It implies that \( A \) satisfies this characteristic equation with roots (or eigenvalues) 2 and 4. Understanding eigenvalues is key as they provide insights into the matrix's characteristics, including its invertibility and its transformation properties.

Non-Singular Matrices

A non-singular matrix, also known as an invertible matrix, is a square matrix that has an inverse. For a matrix \( A \) to be non-singular, it must satisfy a critical property:

• The determinant of \( A \) is not zero.

If \( A \) is non-singular, there exists another matrix \( A^{-1} \) such that:

• \( AA^{-1} = A^{-1}A = I \)

where \( I \) is the identity matrix. In the exercise, the matrix \( A \) is confirmed to be non-singular, implying it is invertible, and we are able to find \( A^{-1} \) during our calculations. The given condition \((A - 2I)(A - 4I) = O\) along with \( A \) being non-singular indicates that neither \( A - 2I \) nor \( A - 4I \) can be the zero matrix, emphasizing the matrix's non-zero eigenvalues.

Characteristic Equation

The characteristic equation of a matrix is a crucial tool in determining the eigenvalues. For a matrix \( A \), the characteristic equation is derived from the polynomial:

• \( \text{det}(A - \lambda I) = 0 \)

where \( \lambda \) represents the eigenvalues, and \( I \) is the identity matrix. This equation helps find the values of \( \lambda \) for which the determinant equals zero, revealing the eigenvalues of \( A \).
In the context of the exercise, we see the characteristic equation in action as \((A - 2I)(A - 4I) = O\), where \( O \) is the zero matrix. This form directly indicates that both 2 and 4 are eigenvalues of the matrix \( A \). The characteristic equation provided reduces to finding which scalars (in this case, 2 and 4) when adjusted by matrix subtraction, then multiplied, results in a zero matrix, strengthening its role in recognizing and utilizing eigenvalues.