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# Determine the dimensions of Nul A and Col A for the matrices shown.$A=\left[ \begin{array}{rrrrrr}{1} & {3} & {-4} & {2} & {-1} & {6} \\ {0} & {0} & {1} & {-3} & {7} & {0} \\ {0} & {0} & {0} & {1} & {4} & {-3} \\ {0} & {0} & {0} & {0} & {0} & {0}\end{array}\right]$

Solution

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Matrix $A$ is already in echelon form so we can conclude that dimension of Col $A$ is 3 as can be seen by the pivot columns.

The matrix is of size $4 \times 6$. Using

$\text{dim (Nul }A) + \text{dim (Col }A) = \text{Number of Columns in }A$

we get:

$\text{dim (Nul }A) + 3 = 6 \Rightarrow \text{dim (Nul }A) = 3$

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