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Explore the rank of a singular 4×4 matrix and its role in determining system consistency and linear transformations.
Contents
Rank of a Singular matrix of order 4, can be at the most
a)1
b)2
c)3
d)4
Answer:
The rank of a singular matrix of order 4 can be at most 4.
Here’s a detailed explanation:
Key Concept:
- The rank of a matrix is the maximum number of linearly independent rows (or columns) it contains. Linearly independent vectors (rows or columns in this case) cannot be expressed as a linear combination of the others.
- A singular matrix is a matrix whose determinant is 0. Intuitively, it means the rows (or columns) are not linearly independent, as if you add rows or columns together in certain ways, you might get the zero vector.
Explanation:
Since a singular matrix of order 4 has a determinant of 0, the rows (or columns) cannot be all linearly independent. However, it’s still possible to have up to 4 linearly independent rows or columns.
Consider the following example:
[[1, 2, 3, 0],
[0, 0, 0, 1],
[2, 4, 6, 0],
[0, 0, 0, 2]]
This matrix has a determinant of 0 (you can calculate it yourself or use software). However, the first, third, and fourth rows are linearly independent (the second row is a linear combination of the first and third). Therefore, the rank of this matrix is 3.
In general, while a singular matrix’s rows or columns cannot all be linearly independent, having up to all 4 be independent is still possible, giving a maximum rank of 4.
Points to Remeber:
- The minimum rank of a singular matrix is 0 (all rows/columns are linearly dependent).
- While the example presented has rank 3, it’s possible to construct singular matrices of order 4 with ranks 1, 2, or 4.
In conclusion, the rank of a singular matrix of order 4 can be at most 4.
What is the Rank of a Singular Matrix?
The rank of a singular matrix is always less than its order (the number of rows and columns). This means:
- For a square matrix (where the number of rows equals the number of columns), the rank will be less than the number of rows/columns.
- For a non-square matrix, the rank will be less than the smaller of the number of rows or columns.
Here’s why:
- A singular matrix has a determinant of 0. This implies that its rows and columns are not linearly independent. In simpler terms, some rows or columns can be obtained by multiplying and adding other rows or columns together.
- The rank of a matrix represents the maximum number of linearly independent vectors in its row space or column space (both are equivalent for square matrices). Since some vectors are dependent in a singular matrix, its rank cannot be the full size of the matrix.
Points To Remember:
- The rank of a singular matrix can be 0 (when all rows/columns are zero) but cannot be equal to its order.
- You can often determine the rank of a singular matrix by performing Gaussian elimination and counting the number of non-zero rows in the echelon form.
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Categories: Math
