Question

1.Various advanced texts in linear algebra prove the following determinant criterion for rank:

The rank of a matrix A is r if and only if A has some r × r sub matrix with a nonzero determinant, and all square sub matrices of larger size have determinant zero.

(A sub matrix of A is any matrix obtained by deleting rows or columns of A. The matrix A itself is also considered to be a sub matrix of A.) Use this criterion to find the rank of the matrix.

1 4 0
3 12 -1
rank (A) =


rank (A) =

Answer 1

$\begin{bmatrix} 1 & 4 & 0\\ 3 & 12 & -1 \end{bmatrix}$ is the given matrix. Performing the operation $R_2 \gets R_2-3R_1$ we get;

$\begin{bmatrix} 1 & 4 & 0\\ 0 & 0 & -1 \end{bmatrix}$ . Now, the sub matrix $\begin{bmatrix} 4 & 0 \\ 0 & -1 \end{bmatrix}$ of order $2\Chi2$ has a non-zero determinant.

Also, this is the largest sub matrix with non-zero determinant.

Thus, rank of the given matrix is 2.