How to Determine Values of a, b, c, d for a Matrix Rank of 2
When analyzing a 4x4 matrix and aiming to reduce its rank to 2, we must identify pairs of linearly dependent rows. This guide will walk you through the process, highlighting the specific values of a, b, c, d required to achieve the desired matrix rank.
Understanding Matrix Rank
In the context of linear algebra, the rank of a matrix is the maximum number of linearly independent rows or columns. For a 4x4 matrix to have a rank of 2, it is crucial to find pairs of linearly dependent rows, meaning that at least two rows can be expressed as linear combinations of the other two.
Identifying Linearly Dependent Rows
To make two rows of the matrix linearly dependent, we need to follow a systematic approach. Let's consider the matrix A:
A begin{pmatrix}a b c d1 0 1 02 0 2 00 1 0 10 2 0 2end{pmatrix}
Rows 1 and 3
For rows 1 and 3 to be linearly dependent, we can set a 0 and b 1. This ensures that the first row is a simple multiple of the third row, allowing for their linear dependence. Let's see the resulting matrix:
A_1 begin{pmatrix}0 1 c d1 0 1 02 0 2 00 1 0 10 2 0 2end{pmatrix}
Rows 2 and 4
Similarly, by setting c 1 and d 0, we can make rows 2 and 4 linearly dependent. This adjustment transforms the matrix into:
A_2 begin{pmatrix}0 1 1 01 0 1 02 0 2 00 1 0 10 2 0 2end{pmatrix}
Reducing the Matrix to Rank 2
With these specific values of a, b, c, d, our matrix can be simplified. Let's perform the necessary row operations to further reduce the matrix to a rank of 2.
We can subtract twice the first row from the third row, resulting in:
R_3 - 2*R_1 2 - 2(0) 2 0 - 2(1) -2 2 - 2(1) 0 0 - 2(0) 0
Similarly, subtract twice the second row from the fourth row:
R_4 - 2*R_2 0 - 2(1) -2 1 - 2(0) 1 0 - 2(1) -2 1 - 2(0) 1
Thus, the matrix becomes:
A finale begin{pmatrix}0 1 1 01 0 1 00 -2 0 00 -2 1 1end{pmatrix}
This final matrix has two zero rows, indicating that its rank is 2.
Conclusion
Determining the values of a, b, c, d for a matrix rank of 2 involves a strategic process of making pairs of rows linearly dependent. By following these steps, we can effectively reduce the rank of a 4x4 matrix to 2. This method is vital in linear algebra and has broader applications in fields like computer graphics, physics, and engineering.