Linear Algebra and Its Applications, Review Exercise 2.21

Review exercise 2.21. Consider an n by n matrix with the value 1 for every entry. What is the rank of such a matrix? Consider another n by n matrix equivalent to a checkerboard, with a_{ij} = 0 if i+j is even and a_{ij} = 1 if i+j is odd. What is the rank of this matrix?

Answer: If each element in the matrix has the value 1 then every column of the matrix is equal to every other column of the matrix, and every row equal to every other row. There is only one column (or row) that is linearly independent (e.g., column 1 or row 1) and thus the rank of the matrix is 1.

For the second matrix, the first and second rows are linearly independent, since the first row has the form (0, 1, 0, 1, \ldots) and the second row has the form (1, 0, 1, 0, \ldots). Every other odd row is equal to the first row: If i is odd then i+j is even whenever 1+j is even, and i+j is odd whenever 1+j is odd; thus we have a_{ij} = a_{1j} for all odd i and all j.

Similarly, every other even row is equal to the second row: If i is even then i+j is even whenever 2+j is even, and i+j is odd whenever 2+j is odd; thus we have a_{ij} = a_{2j} for all even i and all j.

Since the “checkerboard” matrix contains only two linearly independent rows, its rank is 2.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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