Discussion
Find the rank of the given matrix.
\(\begin{bmatrix} 1 & 5 & 6 \\ 2 & 3 & 4 \\ -1 & 2 & 2 \end{bmatrix}\)
Answer: A
The rank of the matrix is 2. First find the determinant: \[ \begin{aligned} &1(3\times 2 - 4\times 2) \\ &- 5(2\times 2 - 4\times(-1)) \\ &+ 6(2\times 2 - 3\times(-1)) \end{aligned} \] \[ = 1(-2) - 5(8) + 6(7) = -2 - 40 + 42 = 0 \] So the rank is less than 3. A 2 by 2 minor is non-zero: \[ \begin{vmatrix} 1 & 5 \\ 2 & 3 \end{vmatrix} = 3 - 10 = -7 \neq 0 \] The rank is therefore 2. The values 4, 6 and 9 are larger than the number of rows and cannot be the rank of a 3 by 3 matrix.
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