The rank of the matrix

 

-4	1	-1
-1	-1	-1
7	-3	1

 is

The correct answer is 2.

To find the rank of the given 3 × 3 matrix, let us denote it as matrix $A$:

Step 1: Calculate the determinant of the 3 × 3 matrix
We first check if the matrix is full rank (rank 3) by calculating its determinant, $\left|A\right|$:

$\left|A\right|=-4·(-1·1-(-1)·(-3\left)\right)-1·(-1·1-(-1)·7)+(-1)·(-1·(-3)-(-1)·7)$

Simplifying the terms inside the parentheses:

$\left|A\right|=-4·(-1-3)-1·(-1+7)-1·(3+7)$

$\left|A\right|=-4·(-4)-1·\left(6\right)-1·\left(10\right)$

$\left|A\right|=16-6-10=0$

Since the determinant of the 3 × 3 matrix is 0, the rank of the matrix must be less than 3.

Step 2: Check for a non-zero 2 × 2 minor
Let us select the submatrix obtained by deleting the third row and third column:

Calculating the determinant of this 2 × 2 minor:

$\left|M_{\mathrm{2\times 2}}\right|=(-4)·(-1)-1·(-1)$

$\left|M_{\mathrm{2\times 2}}\right|=4+1=5\ne 0$

Since there exists at least one 2 × 2 minor with a non-zero determinant, the rank of the matrix is 2.