Question

Deal with the problem of solving \(\mathbf{A x}=\mathbf{b}\) when \(\operatorname{det} \mathbf{A}=0\) Show that \((\mathbf{A x}, \mathbf{y})=\left(\mathbf{x}, \mathbf{A}^{*} \mathbf{y}\right)\) for any vectors \(\mathbf{x}\) and \(\mathbf{y}\)

Short answer

Answer: When the determinant of a matrix is zero, it indicates that the matrix is singular and not invertible. In this case, the system can either have no solution or infinitely many solutions.

Step-by-step solution

Solving \(\mathbf{A x}=\mathbf{b}\) with \(\operatorname{det} \mathbf{A}=0\).

When the determinant of a matrix is zero, it indicates that the matrix is singular and is not invertible. In this circumstance, there can be either no solution or infinitely many solutions to the equation \(\mathbf{Ax}=\mathbf{b}\). The strategy to follow here is to evaluate the augmented matrix \((\mathbf{A} | \mathbf{b})\) and determine the system's properties through row reduction. -Calculate the row echelon form of the augmented matrix \((\mathbf{A} | \mathbf{b})\). -If there is an inconsistency, i.e., a row with all zeros on the left side and a nonzero value on the right, there is no solution to the equation. -If the matrix has a free variable, meaning one or more non-pivot columns, it implies that the system has infinitely many solutions. Express the non-pivot variables in terms of pivot variables for parametric solutions.

Notation and definitions

To prove the equality \((\mathbf{A x}, \mathbf{y})=\left(\mathbf{x}, \mathbf{A}^{*}\mathbf{y}\right)\) for any vectors \(\mathbf{x}\) and \(\mathbf{y}\), let's first define our notation: - \(\mathbf{A}\): an \(m \times n\) matrix - \(\mathbf{x}\): an \(n \times 1\) vector - \(\mathbf{y}\): an \(m \times 1\) vector - \(\mathbf{A^*}\): the adjugate of matrix \(\mathbf{A}\) The inner product of two vectors \(\mathbf{x}\) and \(\mathbf{y}\) is denoted as \((\mathbf{x}, \mathbf{y})\), and the adjugate \(\mathbf{A^*}\) is a matrix that is the transpose of the cofactor of matrix \(\mathbf{A}\).

Calculate both sides of the equation

Now we will calculate \((\mathbf{A x}, \mathbf{y})\) and \((\mathbf{x}, \mathbf{A}^{*}\mathbf{y})\) in order to show they are equal. \((\mathbf{A x}, \mathbf{y})\) can be expanded as follows: $$(\mathbf{A x}, \mathbf{y}) = \sum_{i=1}^m (\mathbf{A x})_i \mathbf{y}_i = \sum_{i=1}^m \left(\sum_{j=1}^n \mathbf{A}_{ij}\mathbf{x}_j\right) \mathbf{y}_i$$ For the other side of the equation: $$(\mathbf{x}, \mathbf{A}^{*}\mathbf{y}) = \sum_{j=1}^n \mathbf{x}_j (\mathbf{A}^{*}\mathbf{y})_j = \sum_{j=1}^n \mathbf{x}_j \left(\sum_{i=1}^m \mathbf{A}^{*}_{ji}\mathbf{y}_i\right)$$

Use the property \(\mathbf{A}^{*}_{ji} = C_{ij}\) of the adjugate

Let's rewrite the second sum using the property that the adjugate elements are the transposed cofactor elements. We will replace \(\mathbf{A}^*_{ji}\) with \(C_{ij}\): $$(\mathbf{x}, \mathbf{A}^{*}\mathbf{y}) = \sum_{j=1}^n \mathbf{x}_j \left(\sum_{i=1}^m C_{ij}\mathbf{y}_i\right)$$

Rearrange the sums

We can switch the order of the two sums in the previous equation: $$(\mathbf{x}, \mathbf{A}^{*}\mathbf{y}) = \sum_{i=1}^m \left(\sum_{j=1}^n C_{ij}\mathbf{x}_j\right) \mathbf{y}_i$$

Use the property \(\sum_{j=1}^n \mathbf{A}_{ij}\mathbf{x}_j = \sum_{j=1}^n C_{ij}\mathbf{x}_j\)

Notice that the sum inside the parentheses is precisely the expansion of the determinant of matrix \(\mathbf{A}\) along the rows. Since the determinant is zero (\(\operatorname{det}\mathbf{A}=0\)), the sum of these products is also zero: $$\sum_{j=1}^n \mathbf{A}_{ij}\mathbf{x}_j = \sum_{j=1}^n C_{ij}\mathbf{x}_j$$

Conclusion

Comparing the expansions of both sides of the equation in steps 3 and 5, and considering the property from step 6, we can conclude that: $$(\mathbf{A x}, \mathbf{y})=\left(\mathbf{x}, \mathbf{A}^{*}\mathbf{y}\right)$$

Key concepts

Singular Matrix

A singular matrix is a square matrix that lacks an inverse. This situation occurs when the determinant of the matrix is zero. The absence of an inverse means you cannot find a unique solution to the matrix equation \( \mathbf{Ax} = \mathbf{b} \). There are two possible outcomes when dealing with a singular matrix in a system of equations:

• No solution: This happens if there is an inconsistency within the system.
• Infinitely many solutions: When the system allows for free variables, suggesting that more than one solution exists.

To tackle these conditions, mathematicians often use techniques like transforming the matrix into Row Echelon Form or utilizing other properties such as the matrix adjugate.

Row Echelon Form

Row Echelon Form is a technique used in linear algebra to simplify matrix calculations. It transforms a matrix into an upper triangular form, aiding in solving systems of linear equations. In this form:

• All non-zero rows are above any rows of all zeros.
• The leading entry of a non-zero row is always to the right of the leading entry in the row above it.

This simplification process allows easy identification of the solutions:

• If a row has leading zeros ending with a non-zero value, the system is inconsistent, meaning there is no solution.
• If any row retains free variables, this indicates infinitely many solutions.

Using Row Echelon Form makes it easier to determine the presence of solutions and the necessity of additional methods for complex equations.

Matrix Adjugate

The adjugate of a matrix is crucial for understanding various properties of matrices. It is formed by taking the transpose of the cofactor matrix of a given matrix \( \mathbf{A} \). This operation follows these steps:

• Calculate the cofactor for each element in the matrix.
• Transpose the resulting cofactor matrix to obtain the adjugate, \( \mathbf{A}^* \).

While a matrix with zero determinant is singular and generally can't be inverted, the adjugate still helps in theoretical analysis, like proving inner product properties or equations involving matrices. For matrices that aren't singular, multiplying the adjugate by the inverse of the determinant yields the inverse of the original matrix.

Inner Product

The inner product is a fundamental concept in linear algebra, representing a type of "dot product" between two vectors. It is defined as:\[(\mathbf{x}, \mathbf{y}) = \sum_{i} x_i y_i\]This operation yields a scalar, offering valuable geometric insights like the angle between vectors and their magnitude. When applied to the context of matrices, creating an inner product between expressions can form equalities useful for broader applications. For instance, showing that \(( \mathbf{Ax}, \mathbf{y}) = ( \mathbf{x}, \mathbf{A}^*\mathbf{y}) \) employs the inner product concept, indicating how matrix operations intertwine with geometric meanings. Understanding this operation is crucial not just for solving equations, but for comprehending relationships amongst vector spaces and other algebraic structures.