Question

Deal with the problem of solving \(\mathbf{A x}=\mathbf{b}\) when \(\operatorname{det} \mathbf{A}=0\) Suppose that det \(\mathbf{A}=0\) and that \(y\) is a solution of \(\mathbf{A}^{*} \mathbf{y}=\mathbf{0} .\) Show that if \((\mathbf{b}, \mathbf{y})=0\) for every such \(\mathbf{y},\) then \(\mathbf{A} \mathbf{x}=\mathbf{b}\) has solutions. Note that the converse of Problem \(27 ;\) the form of the solution is given by Problem \(28 .\)

Short answer

Using the given conditions and the properties of transposes, we showed that if the dot product of b and y is zero for every y that satisfies A*y = 0, then there exists a solution for the equation Ax = b. This is because A*b is equal to the null space of the adjoint of matrix A, implying that a solution for Ax = b exists.

Step-by-step solution

Understanding the problem conditions

Since \(\operatorname{det}\mathbf{A}=0\), it means the matrix \(\mathbf{A}\) is not invertible and \(\mathbf{A}^*\mathbf{y}=\mathbf{0}\). \(\mathbf{A}^*\) is the adjoint of \(\mathbf{A}\). We are given if \((\mathbf{b}, \mathbf{y})=0\) which means the dot product of \(\mathbf{b}\) and \(\mathbf{y}\) is zero for every such \(\mathbf{y}\).

Using given conditions

Let's multiply both sides of the equation \(\mathbf{Ax}=\mathbf{b}\) by \(\mathbf{A}^*\): $$ \mathbf{A}^*\mathbf{Ax} = \mathbf{A}^*\mathbf{b} $$

Applying transpose properties

Using transpose properties, we can rewrite the left side of the equation: $$ (\mathbf{A}^*\mathbf{A})\mathbf{x} = \mathbf{A}^*\mathbf{b} $$ However, since \(\mathbf{A}^*\mathbf{y}=\mathbf{0}\), we know that the left side of the equation must be equal to zero. Therefore, we have the following equation for every such \(\mathbf{y}\) which satisfies \((\mathbf{b}, \mathbf{y})=0\): $$ \mathbf{A}^*\mathbf{b} = \mathbf{0} $$

Finding solutions

From the last equation \(\mathbf{A}^*\mathbf{b} = \mathbf{0}\), we understand that for every such \(\mathbf{y}\) for which \((\mathbf{b}, \mathbf{y})=0\), \(\mathbf{b}\) is in the null space of \(\mathbf{A}^*\). This implies that there exists a solution for \(\mathbf{Ax} = \mathbf{b}\). Thus, we can conclude that if \((\mathbf{b}, \mathbf{y})=0\) for every such \(\mathbf{y}\), the equation \(\mathbf{Ax}=\mathbf{b}\) has solutions.

Key concepts

Adjoint Matrix

An adjoint matrix, denoted as \( \mathbf{A}^* \), is the transpose of the cofactor matrix of \( \mathbf{A} \). It plays a crucial role in matrix equations, especially when dealing with singular matrices where the determinant is zero.

The adjoint matrix helps us understand the solution structure when the original matrix \( \mathbf{A} \) is not invertible. This is because it often appears in calculations involving determinants and inverses. When trying to solve equations where \( \operatorname{det} \mathbf{A} = 0 \), the adjoint can be used to manipulate and simplify expressions, aiding us in finding potential solutions.

It's important to note that even for a non-invertible matrix (where the determinant is zero), the adjoint matrix is always well-defined, providing a way forward in searching for solutions, especially by finding combinations that satisfy both the original and any additional constraints.

In this context, we often check if multiplying by the adjoint matrix can make certain components become zero, which is a strategy used to simplify the analysis of the system of equations.

Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In the context of matrices and systems of equations, the dot product is denoted as \((\mathbf{b}, \mathbf{y})\) and represents the product of corresponding entries of vectors \( \mathbf{b} \) and \( \mathbf{y} \) followed by summation.

For vectors \( \mathbf{b} = [b_1, b_2, ..., b_n] \) and \( \mathbf{y} = [y_1, y_2, ..., y_n] \), the dot product is calculated as:
\[ (\mathbf{b}, \mathbf{y}) = b_1y_1 + b_2y_2 + ... + b_ny_n \]
If the dot product \((\mathbf{b}, \mathbf{y}) = 0\), then the vectors are orthogonal, meaning they are perpendicular to each other in \( \mathbb{R}^n \). This is a significant condition because it suggests certain geometric properties about the vectors.

In our matrix equation problem, having \((\mathbf{b}, \mathbf{y}) = 0\) implies that \( \mathbf{b} \) is orthogonal to any vector \( \mathbf{y} \) from the null space of \( \mathbf{A}^* \), and this orthogonality is key to finding solutions for the equation \( \mathbf{Ax} = \mathbf{b} \).

Null Space

The null space of a matrix is the set of all vectors \( \mathbf{x} \) such that \( \mathbf{A} \mathbf{x} = \mathbf{0} \). It's an essential concept when solving matrix equations, particularly when the matrix is singular and not invertible.

When dealing with the equation \( \mathbf{A} \mathbf{x} = \mathbf{b} \), if \( \mathbf{b} \) is in the null space of \( \mathbf{A}^* \) (the adjoint matrix), it means that multiplying \( \mathbf{A}^* \) with \( \mathbf{b} \) results in the zero vector.

This provides a powerful clue: for solutions to exist to our problem, \( \mathbf{b} \) should be orthogonal to any vector in the null space of \( \mathbf{A}^* \).

Hence, if every solution \( \mathbf{y} \), when used with \( \mathbf{b} \), satisfies the orthogonality condition, it indicates that potential solutions exist for the original system \( \mathbf{Ax} = \mathbf{b} \). The null space helps us identify these conditions by pointing out the vectors that make \( \mathbf{A}^* \mathbf{b} = \mathbf{0} \).

Non-Invertible Matrix

A non-invertible matrix, also known as a singular matrix, is one that does not have an inverse. The condition for a matrix to be invertible is that its determinant is non-zero. Therefore, when \( \operatorname{det} \mathbf{A} = 0 \), \( \mathbf{A} \) is non-invertible.

This property signifies that the matrix cannot be used to directly solve systems of equations by simple inversion, such as \( \mathbf{x} = \mathbf{A}^{-1}\mathbf{b} \).

However, being non-invertible does not always mean that solutions do not exist; rather, it often suggests that solutions may not be unique, or may require special conditions to be identified.

In this scenario, non-invertibility is tackled by considering other approaches like the adjoint matrix, null space, and orthogonal conditions, which can sometimes uncover solutions that are not immediately apparent.

These strategies are instrumental in addressing system equations where directly using an inverse is not possible, ensuring that despite \( \mathbf{A} \) being non-invertible, we can still analyze and solve the equations \( \mathbf{Ax} = \mathbf{b} \) under the given conditions.