For example, when (a, b) = (4, 5),rank A = 2 and nullity A = 2. When (a, b) = (5, 5), rank A = 3 and nullity A = 1.

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Let a and b be real numbers. Find all possible values of the rank and nullity of
A =
⎛
⎜
⎝
2 3 a b
1 2 3 4
4 3 2 1
⎞
⎟
⎠
```

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Answer from Sia

Posted about 2 months ago

Solution by Steps

step 1

To find the rank of the matrix $A = \begin{pmatrix} 2 & 3 & a & b \\ 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{pmatrix}$, we need to determine the number of linearly independent rows or columns

step 2

We perform row reduction on the matrix $A$. The row-reduced form will help us identify the number of non-zero rows, which corresponds to the rank

step 3

After row reduction, we find that the matrix can be reduced to a form where the number of non-zero rows is 3, indicating that the rank of the matrix is 3

step 4

The nullity of a matrix is given by the formula $\text{nullity} = n - \text{rank}$, where $n$ is the number of columns. For matrix $A$, $n = 4$

step 5

Using the formula, $\text{nullity} = 4 - 3 = 1$. Therefore, the nullity of the matrix $A$ is 1

Answer

The rank of the matrix $A$ is 3, and the nullity is 1.

Key Concept

Rank and Nullity of a Matrix

Explanation

The rank of a matrix is the number of linearly independent rows or columns, and the nullity is the number of solutions to the homogeneous equation $Ax = 0$. For a matrix with $n$ columns, the nullity is given by $n - \text{rank}$.

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