# Question Video: Finding the Rank of a 3 Γ 3 Matrix Using Determinants Mathematics

Find the rank of the following matrix: [1, 2, 4 and 7, β1, β2 and 2, 4, 8].

02:22

### Video Transcript

Find the rank of the following matrix.

Recall that the rank of a matrix π΄ is the number of rows or columns in the largest square submatrix of π΄ with a nonzero determinant. Recall also that the rank of π΄ is greater than or equal to zero and less than or equal to the minimum of π and π, where π is the number of rows in π΄ and π is the number of columns in π΄. Since π΄ in this case is a three-by-three matrix, the rank of π΄ is between zero and three. Recall also that the rank of π΄ is equal to zero if and only if π΄ is the zero matrix. This matrix clearly isnβt the zero matrix. Therefore, its rank cannot be zero.

The largest possible square submatrix of this matrix 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 zero. This is the only possible three-by-three submatrix of π΄, and it has a determinant of zero. Therefore, the rank of π΄ cannot be three. We now seek a two-by-two submatrix of π΄ with a nonzero determinant. This is a problem because there are nine possible two-by-two submatrices of π΄, and we may need to check every single one of them.

Consider, for instance, if the original matrix had looked like this. The only two-by-two submatrix we can select that might have a nonzero determinant is this one. In this hypothetical example, the choice is clear, but it might not be for our question. If we look at the original matrix, we can see that the bottom row is an exact scalar multiple of the top row. Any two-by-two submatrix selected from these two rows will have determinant of zero. And we might suppose from this that this means that there are no two-by-two submatrices with a nonzero determinant.

We might suppose from this that there are no two-by-two submatrices of π΄ with a nonzero determinant. However, if we select a two-by-two submatrix that doesnβt come from just these two rows that are scalar multiples of each other, for example, by removing the bottom row and the right-most column, we get a nonzero determinant. We have therefore found a two-by-two submatrix of π΄ with a nonzero determinant. Therefore, the rank of π΄ is two.

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