Video Transcript
Find the rank of the matrix
negative 16, negative 11, negative 14, 17, 19, negative 24, three, negative six,
negative 24.
Recall that the rank of a matrix 𝐴
is the number of rows or columns in the largest square submatrix of 𝐴 with a
nonzero determinant. Recall also that the rank of 𝐴 is
greater than or equal to zero and less than or equal to the minimum of 𝑝 and 𝑞,
where 𝑝 is the number of rows in 𝐴 and 𝑞 is the number of columns in 𝐴. This is a three-by-three
matrix. The rank of 𝐴 must therefore be
between zero and three. And finally, recall that the rank
of 𝐴 is equal to zero if and only if 𝐴 is the zero matrix. This matrix clearly isn’t the zero
matrix. Therefore, its rank cannot be
zero.
The largest possible square
submatrix of 𝐴 is just itself, a three-by-three matrix. Taking the determinant of the
matrix by expanding along the top row, we get a result of 8130, which is not equal
to zero. We have therefore found a
three-by-three submatrix of 𝐴, in this case itself, with a nonzero determinant. Therefore, the rank of 𝐴 is
three.
