Problem 6 - Differential Equations and Linear Algebra

Question

Determine the nullity of \(A\) "by inspection" by appealing to the Rank-Nullity Theorem. Avoid computations. $$A=\left[\begin{array}{rr}2 & -3 \\\0 & 0 \\\\-4 & 6 \\\22 & -33\end{array}\right]$$

Step-by-Step Solution

Answer

By observing the given matrix A, we notice that the third and the fourth rows are linearly dependent on the first row, and the second row is a row of zeros. Hence, the rank of matrix A is 1. Using the Rank-Nullity Theorem, we find that the nullity of A is 1.

Step 1: Observe the given matrix A

Carefully observe the following matrix A: $$A=\left[\begin{array}{rr}2 & -3 \\\0 & 0 \\\-4 & 6 \\\22 & -33\end{array}\right]$$ Notice the third row, which is -2 times the first row, and the fourth row, which is 11 times the first row, i.e, $$Row_3 = -2 \times Row_1$$ $$Row_4 = 11 \times Row_1$$

Step 2: Determine the rank of matrix A

Since the third and the fourth row are linearly dependent on the first row, they do not contribute to the rank. The second row is a row of zeros, which also does not contribute to the rank of A. Therefore, the rank of matrix A is 1, i.e., Rank(A) = 1

Step 3: Use the Rank-Nullity Theorem to find the nullity

Now that we have the rank of matrix A, we can apply the Rank-Nullity Theorem: Rank(A) + Nullity(A) = number of columns in matrix A (in this case, 2) Substitute the values we have: 1 + Nullity(A) = 2 Solve for Nullity(A): Nullity(A) = 2 - 1

Step 4: Nullity of matrix A

Finally, after solving the equation for Nullity(A), we have: Nullity(A) = 1 Thus, the nullity of the given matrix A "by inspection" is 1.

Key Concepts

Understanding Linear Dependence

Linear dependence is a crucial concept in linear algebra. It helps determine the relationships between rows or columns in a matrix. When rows (or columns) are linearly dependent, it means that some rows can be expressed as combinations of others.

In the given matrix \(A\), both the third and fourth rows are linearly dependent on the first row. Specifically:

• The third row is \(-2\) times the first row.
• The fourth row is \(11\) times the first row.

Hence, these dependent rows do not add to the matrix's rank, affecting how we calculate it. Recognizing linear dependence allows us to simplify the matrix analysis, focusing on independent rows that contribute to the rank.

Understanding Matrix Rank

The rank of a matrix is the count of its linearly independent rows or columns. It's a fundamental property, providing insight into the solutions of linear systems.

For the matrix \(A\), since the third and fourth rows are dependent on the first, the rank depends on the independent rows. The second row is all zeros, which also doesn't contribute to the rank. Therefore, we only consider the first row here, which is independent, resulting in:

• Rank of \(A = 1\)

The rank helps us understand how many of the original vector space's dimensions are spanned by the matrix.

Exploring the Nullity of a Matrix

Nullity is a concept closely tied to the Rank-Nullity Theorem. It measures the dimension of the solution space to the homogeneous equation \(Ax = 0\).

The Rank-Nullity Theorem states:\[\text{Rank}(A) + \text{Nullity}(A) = \text{Number of Columns in } A\]In this case, since the rank is 1 and the matrix has 2 columns, we have:

• Rank = 1
• Columns = 2

Using the theorem, we calculate nullity as follows:\[1 + \text{Nullity}(A) = 2\]\[\text{Nullity}(A) = 2 - 1 = 1\]This result tells us that there is 1 independent direction in the solution space of \(Ax = 0\), despite the matrix having a rank of 1.