Problem 21 - Algebra for JEE ADVANCED

Question

If \(P\) is an orthogonal matrix and \(Q=P A P^{T}\) and \(x=P^{T} Q^{1000} P\), then \(x^{-1}\) is, where \(A\) is involutary matrix a. \(A\) b. \(I\) c. \(A^{1000}\) d. none of these

Step-by-Step Solution

Answer

The inverse of x is A, option (a).

Step 1: Understanding the Orthogonal Matrix

An orthogonal matrix, by definition, satisfies the property that its transpose is also its inverse: \[ P^{T} = P^{-1} \] This property will be important in simplifying the expressions involving products of these matrices.

Step 2: Utilizing Matrix Properties

Given that matrix \(A\) is involutory, we have:\[ A = A^{-1} \] This means that \(A^2 = I\), where \(I\) is the identity matrix.

Step 3: Simplifying Matrix Q

The expression for matrix \(Q\) is given as:\[ Q = P A P^{T} \] Here, because \(P\) is orthogonal, we know \(P^{T} = P^{-1}\), so \(Q\) seems like it rearranges \(A\), but \(P\) and \(P^{T}\) simplify out in future steps.

Step 4: Simplifying Repeated Multiplication and Inverse

To compute \(x = P^{T} Q^{1000} P\), first recognize:\[ Q^{1000} = (P A P^{T})^{1000} \]Using properties of powers, we see that this is equivalent to:\[ Q^{1000} = P A^{1000} P^{T} \] Therefore, substituting back,\[ x = P^{T} (P A^{1000} P^{T}) P = A^{1000} \]This simplifies because \(P^{T} P\) cancels out as the identity matrix.

Step 5: Finding the Inverse of x

Since we have \(x = A^{1000}\) and knowing \(A\) is involutory with \(A^2 = I\), any even power \(A^{2n}=I\). Hence, \(A^{1000}\) is simply the prior inverse, corresponding here to:\[ x^{-1} = A^{-1000} = (A^{1000})^{-1} = A \] Since each \(A\) is its own inverse, and even squares resolve to the identity matrix.

Key Concepts

Involutory Matrices

Involutory matrices are unique types of matrices that hold a special property: when you multiply them by themselves, you end up with the identity matrix. In mathematical terms, if a matrix \( A \) is involutory, then \( A^2 = I \), where \( I \) is the identity matrix.

This means:

• An involutory matrix is its own inverse, i.e., \( A^{-1} = A \).
• You can raise an involutory matrix to any even power and it will become the identity matrix: \( A^{2n} = I \).

Because of these properties, involutory matrices are particularly interesting in problems involving matrix powers and inverses. They make certain complex calculations simpler, as the matrix effectively "cancels itself out" when squared or raised to any even power.

In our exercise, matrix \( A \) being involutory means that \( A^{1000} \) returns to \( I \), simplifying the problem significantly.

Matrix Inverse

The concept of a matrix inverse is similar to the inverse in arithmetic. For a square matrix \( B \), its inverse is another matrix \( B^{-1} \) such that when \( B \) is multiplied by \( B^{-1} \), the result is the identity matrix:

• \( B B^{-1} = I \)
• \( B^{-1} B = I \)

The existence of an inverse depends on the determinant of the matrix; if the determinant is non-zero, the inverse exists. An interesting feature of involutory matrices is that each is its own inverse. Thus, understanding involutory matrices helps reinforce general concepts related to matrix inverses.

In the exercise, simplifying \( x^{-1} = A^{1000} \), where \( A^{-1} = A \), shows how these properties ensure \( A \) resolves again as the inverse.

Identity Matrix

The identity matrix, denoted as \( I \), is a fundamental building block in matrix algebra, akin to the number 1 in real number multiplication. It has 1s on the diagonal from upper left to lower right and zeros elsewhere. Multiplying any matrix \( C \) by the identity matrix results in the same matrix \( C \). Some crucial aspects include:

• It acts as a multiplicative identity: \( C \, I = I \, C = C \).
• Identity matrices are instrumental in solving equations involving matrices, allowing for simplifications when dealing with powers or compositions of matrices.

In the original problem, the identity matrix plays a significant role because it appears when involutory matrices are squared. Understanding how the identity matrix stays consistent through matrix operations, even when simplified drastically in cases like involutory matrices, is key to solving complex matrix equations efficiently. This was used in simplifying the expression for \( x \) in the problem statement.