Question Video: Finding the Rank of a Given Matrix Mathematics

Find the rank of the matrix [−16, −11, −14 and 17, 19, −24 and 3, −6, −24].


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.

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