Problem 63 - Algebra for JEE ADVANCED

Question

If \(A^{2}-A+I=0\), then the inverse of \(A\) is a. \(A^{-2}\) b. \(A+I\) c. \(I-A\) d. \(A-I\)

Step-by-Step Solution

Answer

The inverse of \( A \) is \( I - A \), option c.

Step 1: Understand the given equation

We are given the equation \( A^2 - A + I = 0 \). Rearrange this equation to isolate \( A^2 \): \[ A^2 = A - I \].

Step 2: Solve for the inverse of A

To find the inverse of \( A \), we multiply both sides of \( A^2 = A - I \) by \( A^{-1} \), assuming \( A \) is invertible. Thus, \( A A^{-1} = I \) and \( A^2 A^{-1} = (A - I)A^{-1} \). Simplify: \[ A = I - A^{-1} \].

Step 3: Verify the expression for inverse

From \( A = I - A^{-1} \), rearrange to find: \( A^{-1} = I - A \). Thus, the inverse of \( A \) is \( I - A \) which matches option c. \( A^{-1} = I - A \).

Key Concepts

Matrix Algebra

Matrix Algebra is a fundamental area of mathematics that deals with arrays of numbers, called matrices, and operations that can be performed on them. Matrices are used to solve systems of linear equations, among other applications, and clearly play a crucial role in various fields like physics, computer graphics, and statistics.

Some of the basic operations in matrix algebra include:

• Addition and Subtraction: Matrices of the same dimension can be added or subtracted by summing or subtracting corresponding elements.
• Multiplication: This involves dot product operations between rows and columns. Note that matrix multiplication is not commutative, which means that the order of the matrices matters.
• Scalar Multiplication: Every element of the matrix is multiplied by the scalar.
• Identity Matrix: Denoted as \( I \), this matrix has 1s on the diagonal and 0s elsewhere, acting as an "unchangeable element" in multiplication.

Understanding these basic operations helps in solving more complex problems such as finding matrix inverses, solving matrix equations, and determining eigenvalues and eigenvectors.

Inverse of a Matrix

The inverse of a matrix \( A \) is a matrix \( A^{-1} \) such that when you multiply them together, you get the identity matrix, \( I \). Not all matrices have inverses, and a matrix must be square (same number of rows and columns) and have a non-zero determinant to be invertible.

To calculate the inverse of a 2x2 matrix, for instance:

• Swap the elements of the leading diagonal.
• Change the sign of the off-diagonal elements.
• Multiply each term by \( 1/\text{det}(A) \), where \( \text{det}(A) \) is the determinant.

The process is more complex for larger matrices but follows similar principles involving adjugates and determinants.

In the exercise, by rearranging and manipulating the equation \( A^2 = A - I \), it is shown that the inverse \( A^{-1} \) is \( I - A \). This is an exemplification of how matrix algebra is applied to find inverses in linear algebra.

Linear Algebra

Linear Algebra is a branch of mathematics that deals with vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. It is the framework upon which we study matrix algebra and matrix inversion.

Key concepts in linear algebra include:

• Vector Spaces: A collection of vectors that can be scaled and added together to form new vectors.
• Linear Transformations: Functions that map one vector space to another while preserving vector addition and scalar multiplication.
• Systems of Linear Equations: Represented and solved using matrices, relevant for various applications in science and engineering.
• Determinants and Eigenvalues: Properties of matrices used to explore solutions and transformations in vector spaces.

Understanding how to manipulate and solve equations using matrices, as illustrated in the problem, is fundamental in linear algebra. Learning these principles enables solving complex problems, such as determining the intersections of planes, performing transformations in computer graphics, and optimizing resources in operations research.