Understanding the Nontrivial Solutions of Homogeneous Equations for Overdetermined Matrices with Two Pivot Positions in Linear Algebra

If A is 3×2 with two pivot positions, then Ax = 0 has a nontrivial solution.

To understand why this statement is true, let’s first define what is meant by a pivot position in the context of a matrix

To understand why this statement is true, let’s first define what is meant by a pivot position in the context of a matrix.

In linear algebra, a pivot position refers to a position in a matrix where the leading entry (the first nonzero entry) of a row occurs. In other words, it is the entry that determines the row’s pivoting behavior during row reduction or the Gaussian elimination process.

Now, let’s consider matrix A, which is a 3×2 matrix. This means that A has 3 rows and 2 columns. Since A has more rows than columns, it is an overdetermined system.

Given that A has two pivot positions, it means that during row reduction, only two rows of A will have leading entries. Let’s say that the rows with pivot positions are row 1 and row 2.

Now, let’s consider the homogeneous equation Ax = 0, where x is a column vector of variables.

Since A is a 3×2 matrix and x is a 2×1 column vector, the equation Ax = 0 represents a system of three linear equations:

a11 * x1 + a12 * x2 = 0 (equation 1)
a21 * x1 + a22 * x2 = 0 (equation 2)
a31 * x1 + a32 * x2 = 0 (equation 3)

Here, a11, a12, a21, a22, a31, and a32 are the entries of matrix A.

Now, because rows 1 and 2 have pivot positions, we know that at least two of the equations (equation 1 and equation 2) will have leading entries. The presence of pivot positions means that these leading entries cannot be zero.

Let’s consider the case where only equations 1 and 2 have leading entries. This implies that a11 and a21 cannot be zero (since they are the leading entries). If a11 and a21 are not zero, we can express equation 1 and equation 2 as:

a11 * x1 + a12 * x2 = 0 (equation 1′)
a21 * x1 + a22 * x2 = 0 (equation 2′)

Now, let’s solve equations 1′ and 2′ simultaneously. By doing so, we can express x2 in terms of x1:

x2 = (-a11 / a12) * x1 (equation 4′)

As you can see, equation 4′ represents a nontrivial solution. It means that there exists a non-zero solution for x when A is a 3×2 matrix with two pivot positions.

In general, if a matrix has more rows than columns and at least one pivot position in each row, the homogeneous equation Ax = 0 will always have a nontrivial solution. This is because there will always be variables that can be expressed in terms of other variables.

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