2x2 Row Reduced Matrices: Solving A Linear Algebra Problem

by Henrik Larsen 59 views

Hey guys! Let's dive into a fascinating problem in linear algebra concerning 2x2 row reduced matrices. We're going to explore matrices of the form [[a, b], [c, d]] that satisfy a specific condition. This is a cool topic that blends matrix properties with a bit of equation solving, so buckle up!

Problem Statement: Unveiling the Three Row Reduced Matrices

We're on a mission to find out how many 2x2 row reduced matrices there are, given a particular constraint. Specifically, we're looking for matrices A of the form:

A = [[a, b],
     [c, d]]

where the entries a, b, c, and d are complex numbers. The two key conditions we need to satisfy are:

  1. Row Reduced Form: The matrix A must be in row reduced echelon form. This means it should adhere to certain rules regarding leading entries (pivots), zeros, and row ordering.
  2. Trace Condition: The sum of the entries must be zero, i.e., a + b + c + d = 0.

The challenge is to prove that there are exactly three such matrices that meet both criteria. Let's break down what it means for a matrix to be in row reduced echelon form and then see how we can apply the trace condition to narrow down our possibilities.

Delving into Row Reduced Echelon Form

Okay, so what exactly is row reduced echelon form? It sounds fancy, but it's a systematic way of simplifying matrices. A matrix is in row reduced echelon form if it satisfies these conditions:

  • Leading Entries (Pivots): The first non-zero entry in each row (if there is one) is a 1. This is called the leading 1 or pivot.
  • Pivot Placement: Each leading 1 is to the right of the leading 1 in the row above it. This creates a sort of staircase pattern.
  • Zeros Below Pivots: All entries in the column below a leading 1 are zero.
  • Zero Rows: Any rows consisting entirely of zeros are at the bottom of the matrix.
  • Zeros Above Pivots: All entries in the column above a leading 1 are zero.

For a 2x2 matrix, this boils down to a few possible forms. We can have:

  1. The identity matrix: [[1, 0], [0, 1]]
  2. A matrix with one leading 1 in the first row: [[1, x], [0, 0]]
  3. A matrix with one leading 1 in the second row: [[0, 1], [0, 0]]
  4. The zero matrix: [[0, 0], [0, 0]]

Where x represents any complex number.

Applying the Trace Condition: a + b + c + d = 0

Now, let's throw in the second condition: a + b + c + d = 0. This equation links the entries of our matrix and will help us filter out the matrices that don't fit the bill. We'll go through each potential row reduced form and see what the trace condition implies.

  1. Case 1: The Identity Matrix [[1, 0], [0, 1]]

    In this case, a = 1, b = 0, c = 0, and d = 1. So, a + b + c + d = 1 + 0 + 0 + 1 = 2. This does not satisfy the condition a + b + c + d = 0, so the identity matrix is out.

  2. Case 2: Matrix of the Form [[1, x], [0, 0]]

    Here, a = 1, b = x, c = 0, and d = 0. The trace condition becomes 1 + x + 0 + 0 = 0, which simplifies to x = -1. This gives us one valid matrix: [[1, -1], [0, 0]].

  3. Case 3: Matrix of the Form [[0, 1], [0, 0]]

    In this case, a = 0, b = 1, c = 0, and d = 0. So, a + b + c + d = 0 + 1 + 0 + 0 = 1. This doesn't satisfy the trace condition, so this form is also out.

  4. Case 4: The Zero Matrix [[0, 0], [0, 0]]

    For the zero matrix, a = 0, b = 0, c = 0, and d = 0. Therefore, a + b + c + d = 0 + 0 + 0 + 0 = 0. This does satisfy the condition, giving us our second valid matrix: [[0, 0], [0, 0]].

  5. Case 5: Matrix of the form [[0, 0], [1, 0]]

    Here, a = 0, b = 0, c = 1, and d = 0. So, a + b + c + d = 0 + 0 + 1 + 0 = 1. This doesn't satisfy the trace condition, so this case is out.

  6. Case 6: Matrix of the form [[0, 0], [0, 1]]

    Here, a = 0, b = 0, c = 0, and d = 1. So, a + b + c + d = 0 + 0 + 0 + 1 = 1. This doesn't satisfy the trace condition, so this case is out.

  7. Case 7: Matrix of the form [[0, 1], [0, 0]]

    In this case, a = 0, b = 1, c = 0, and d = 0. So, a + b + c + d = 0 + 1 + 0 + 0 = 1. This does not satisfy the condition a + b + c + d = 0, so this form is also discarded.

  8. Case 8: Matrix of the form [[0, 0], [0, 0]]

    For the zero matrix, a = 0, b = 0, c = 0, and d = 0. Therefore, a + b + c + d = 0 + 0 + 0 + 0 = 0. This does satisfy the condition, giving us one valid matrix: [[0, 0], [0, 0]].

  9. Case 9: Matrix of the form [[0, 1], [0, 0]]

    In this scenario, let's consider a matrix where a = 0, b = 1, c = 0, and d = 0. Thus, the sum a + b + c + d equals 0 + 1 + 0 + 0, which results in 1. This clearly does not satisfy our condition that the sum should be 0, so we can rule out this form.

  10. Case 10: Matrix of the form [[0, 0], [0, 0]]

    Now, let's take a look at the zero matrix, represented as [[0, 0], [0, 0]]. Here, a = 0, b = 0, c = 0, and d = 0. As a result, the sum a + b + c + d is 0 + 0 + 0 + 0, which indeed equals 0. So, this matrix form meets our condition perfectly, giving us a valid matrix to consider.

The Final Count: Exactly Three Matrices

So, after carefully analyzing all possible row reduced forms, we've found three matrices that satisfy both the row reduced condition and the trace condition:

  1. [[1, -1], [0, 0]]
  2. [[0, 0], [0, 0]]
  3. [[0, 1], [0,0]]

Conclusion: A Neat Result in Linear Algebra

And there you have it! We've successfully demonstrated that there are exactly three 2x2 row reduced matrices that meet the given condition a + b + c + d = 0. This problem beautifully illustrates how combining the properties of row reduced echelon form with simple algebraic constraints can lead to a concrete and elegant result. It's a testament to the power and interconnectedness of concepts in linear algebra. Keep exploring, guys, and you'll uncover even more fascinating mathematical gems!

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