Find the rank of the matrix two,
24, four, 48.
Recall that the rank of a matrix 𝐴
is the number of rows or columns 𝑛 of the largest square 𝑛-by-𝑛 submatrix of 𝐴
with a nonzero determinant. This implies that for a two-by-two
matrix like the one we have, the rank is between zero and two. Recall also that the rank of 𝐴 is
equal to zero if and only if 𝐴 is the zero matrix. Clearly, this matrix is not the
zero matrix. Therefore, its rank is not equal to
zero. Taking the determinant of the
original matrix, we get two times 48 minus 24 times four, which is equal to
zero. Since the only two-by-two submatrix
of 𝐴 is itself, there is no two-by-two submatrix with a nonzero determinant. Therefore, the rank of 𝐴 cannot be
two. And this leaves only one option
remaining: the rank of 𝐴 must be equal to one.