Question Video: Finding the Rank of a 2 × 3 Matrix Using Determinants Mathematics

Video Transcript

Find the rank of the following matrix using determinants: seven, six, eight, negative eight, three, eight.

Recall that the rank of a matrix 𝐴 is the number of rows or columns of the largest square 𝑛-by-𝑛 submatrix of 𝐴 with a nonzero determinant. Recall also that the rank of the matrix is between zero and the minimum of 𝑝 and 𝑞, where 𝑝 is the number of rows of 𝐴 and 𝑞 is the number of columns of 𝐴. This matrix has two rows and three columns. Therefore, the rank of 𝐴 must be less than or equal to the smaller of these numbers, which is two. Recall also 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.

We now seek the largest square submatrix of the original matrix with a nonzero determinant. The largest possible square submatrix of the original matrix will be a two by two. So let’s choose the two-by-two matrix formed from deleting the right-most column. Taking the determinant of this submatrix, we get seven times three minus six times negative eight, which is equal to 21 plus 48, which is equal to 69, which is not equal to zero. We have therefore found a two-by-two submatrix of the original matrix with a nonzero determinant. Therefore, the rank of the original matrix is two.