Question:

Let \( L \), \( M \), and \( N \) be non-singular matrices of size \( 3 \times 3 \), such that \( L^2 = L^{-1} \), \( M = L^8 \), and \( N = L^2 \). Find \( |M - N| \).

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When dealing with matrix powers and inverses, it's often helpful to use properties like \( L^4 = I \) and matrix identities to simplify the calculations.

Updated On: Feb 14, 2025

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The Correct Option is A

Solution and Explanation

Step 1: Using the given conditions. We are given that \( L^2 = L^{-1} \), so: \[ L^4 = I \quad \text{(since multiplying both sides by \( L^2 \))} \] Step 2: Calculate \( M - N \). From the given relations: \[ M = L^8 = (L^4)^2 = I^2 = I \] \[ N = L^2 \] Thus, \( M - N = I - L^2 \). Step 3: Finding the determinant. Since \( L^2 = L^{-1} \), we have: \[ I - L^2 = 0 \quad \text{(because \( L^2 \) is the inverse of \( L \))} \] Hence, the determinant of \( M - N \) is \( 0 \): \[ |M - N| = 0 \] Thus, the correct answer is \( 0 \).

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Questions Asked in GATE CS exam

Consider the following code:

int a;
int arr[] = {30, 50, 10};
int *ptr = arr[10] + 1;
a = *ptr;
(*ptr)++;
ptr = ptr + 1;
printf("%d", a + arr[1] + *ptr);

GATE CS - 2025
Trees

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What is the output of the following C code?

  void foo(int *p, int x) {
      *p = x;
  }

  void main() {
      int *z;
      int a = 20, b = 25;
      z = a;  // Incorrect: Should be z = a;
      foo(z, b);
      printf("%d", a);
  }

Issue: The statement z = a; is invalid because a is an integer, and z is a pointer.

GATE CS - 2025
Trees