Problem 37 - Algebra for JEE ADVANCED

Question

If \(A\) and \(B\) are two non-singular matrices of the same order such that \(B^{r}=I\), for some positive integer \(r>1\). Then \(A^{-1} B^{r-1} A-A^{-1} B^{-1} A=\) a. \(I\) b. \(2 l\) c. \(O\) d. \(-I\)

Step-by-Step Solution

Answer

The answer is c. \(O\).

Step 1: Understand the Problem

We need to find the difference between two expressions involving matrix operations: \(A^{-1} B^{r-1} A\) and \(A^{-1} B^{-1} A\). Given that \(B^r = I\), our task involves simplifying these terms and finding their difference.

Step 2: Simplify and Analyze Given Information

From the problem, we know that \(B^r = I\), which implies that \(B\) raised to the power of \(r\) equals the identity matrix \(I\). This is key to simplifying the terms involving \(B\).

Step 3: Express \(B^{r-1}\) and \(B^{-1}\)

Since \(B^r = I\), then \(B^{r-1} \times B = I\). Therefore, \(B^{-1} = B^{r-1}\). This tells us that the expressions \(A^{-1} B^{r-1} A\) and \(A^{-1} B^{-1} A\) actually simplify to the same matrix operations.

Step 4: Calculate the Difference

The expression \(A^{-1} B^{r-1} A - A^{-1} B^{-1} A\) simplifies to \(A^{-1} B^{r-1} A - A^{-1} B^{r-1} A\), since \(B^{-1} = B^{r-1}\). This results in the zero matrix, \(O\).

Step 5: Choose the Correct Option

The correct option that corresponds to a result of the zero matrix is c. \(O\).

Key Concepts

Non-singular Matrices

In matrix algebra, a non-singular matrix is one that has an inverse. But what does that mean? In simpler terms, if you can find another matrix such that multiplying the two matrices together results in the identity matrix, then the first matrix is known as non-singular. This property is crucial because it ensures that the matrix can "undo" its effects via multiplication, just like multiplying a number by its reciprocal results in one.

• Non-singular matrices have a non-zero determinant.
• The existence of an inverse makes certain algebraic manipulations possible, such as solving systems of linear equations.
• Matrices that are not non-singular are called singular and do not have an inverse.

Understanding non-singular matrices is fundamental for progressing into more advanced matrix operations like matrix inversion.

Matrix Inversion

Matrix inversion involves finding a matrix that, when multiplied by the original matrix, yields the identity matrix. If matrix \(A\) is non-singular, then its inverse, denoted \(A^{-1}\), satisfies the equation \(AA^{-1} = I\).

• Matrix inversion is analogous to finding reciprocals in arithmetic.
• Not all matrices can be inverted. Only non-singular matrices have inverses.
• Inversion is widely used to solve linear equations where the coefficient matrix is non-singular.

Learning matrix inversion enables solving complex mathematical problems, including systems of equations efficiently.

Identity Matrix

The identity matrix functions as the multiplicative identity in matrix operations, similar to how the number one functions in multiplication for real numbers.
Each identity matrix is a square matrix, with ones on the diagonal and zeros elsewhere; denoted as \(I\). For example, a 3x3 identity matrix looks like this:
\[I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{bmatrix}.\]

• Multiplying any matrix by the identity matrix results in the original matrix.
• In matrix algebra, the role of the identity matrix is pivotal, especially in matrix inversion and proof.
• The identity matrix remains unchanged when multiplied by another matrix, preserving the properties of the original matrix.

Matrix Subtraction

Matrix subtraction is a basic operation in matrix algebra that involves subtracting corresponding elements from two matrices of the same dimension.
Here’s a simple explanation to help: if you have matrices \(C\) and \(D\), both with elements arranged the same way, then \(C - D\) results in another matrix where each element is the difference of the corresponding elements in \(C\) and \(D\).

• Matrices must be of the same dimension to perform subtraction.
• Subtraction is remarkably similar to addition, with each element being subtracted individually.
• Useful in contexts like solving matrix equations and adjusting weights in machine learning models.

Subtraction in matrices shares similarities with scalar subtraction, making it an easy concept once mastered.