Problem 42 - Advanced Engineering Mathematics

Question

Consider the \(3 \times 3\) diagonal matrix $$ \mathbf{A}=\left(\begin{array}{rrr} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \end{array}\right) . $$ Determine conditions such that \(\mathbf{A}\) is nonsingular. If \(\mathbf{A}\) is nonsingular, find \(\mathbf{A}^{-1}\). Generalize your results to an \(n \times n\) diagonal matrix.

Step-by-Step Solution

Answer

\(\mathbf{A}\) is nonsingular if none of its diagonal elements are zero. Its inverse is obtained by reciprocating each diagonal element.

Step 1: Understanding Nonsingular Matrix

A matrix is nonsingular if its determinant is not equal to zero. For a diagonal matrix like \(\mathbf{A}\), the determinant is the product of its diagonal elements: \[det(\mathbf{A}) = a_{11} \cdot a_{22} \cdot a_{33}.\] For \(\mathbf{A}\) to be nonsingular, this product must not be zero, meaning none of the diagonal elements are zero.

Step 2: Checking Conditions for Nonsingularity

To ensure \(\mathbf{A}\) is nonsingular, the conditions are: \(a_{11} eq 0\), \(a_{22} eq 0\), and \(a_{33} eq 0\). These conditions guarantee that the determinant of \(\mathbf{A}\) is not zero.

Step 3: Finding the Inverse of a Nonsingular Matrix \(\mathbf{A}\)

For a diagonal matrix, the inverse is obtained by taking the reciprocal of each non-zero diagonal element. Thus, if \(\mathbf{A}\) is nonsingular, its inverse, \(\mathbf{A}^{-1}\), is given by: \[\mathbf{A}^{-1} = \begin{pmatrix} 1/a_{11} & 0 & 0 \ 0 & 1/a_{22} & 0 \ 0 & 0 & 1/a_{33} \end{pmatrix}.\]

Step 4: Generalizing to an \(n \times n\) Diagonal Matrix

For a general \(n \times n\) diagonal matrix \(\mathbf{B}\) with elements \(b_{ii}\) on its diagonal, \(\mathbf{B}\) is nonsingular if each \(b_{ii} eq 0\). If \(\mathbf{B}\) is nonsingular, its inverse is:\[\mathbf{B}^{-1} = \begin{pmatrix} 1/b_{11} & 0 & \cdots & 0 \ 0 & 1/b_{22} & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1/b_{nn} \end{pmatrix}.\]

Key Concepts

Nonsingular Matrix

A matrix is termed 'nonsingular' when it possesses an inverse. This is possible only when its determinant is not equal to zero. For diagonal matrices, this condition simplifies greatly. All elements along the diagonal must be non-zero. Thus, a matrix

• has a non-zero determinant when
• each of its diagonal elements is non-zero.

In the case of a diagonal matrix \[ \mathbf{A} = \begin{pmatrix} a_{11} & 0 & 0 \ 0 & a_{22} & 0 \ 0 & 0 & a_{33} \end{pmatrix} \]its determinant is simply the product of its diagonal elements: \[ det(\mathbf{A}) = a_{11} \cdot a_{22} \cdot a_{33}. \]Ensure each element \(a_{11}, a_{22}, a_{33}\) is non-zero to maintain a nonsingular status. This rule holds for the nonsingularity of any diagonal matrix, even in higher dimensions (\(n \times n\) matrices). All diagonal elements must be non-zero for the matrix to be nonsingular.

Matrix Inversion

If a matrix is nonsingular, it has an inverse, often denoted as \(\mathbf{A}^{-1}\). For diagonal matrices, finding the inverse is a straightforward process. Since these matrices have zeroes everywhere except on the diagonal, the inverse involves taking reciprocals of those non-zero diagonal entries. Consider a nonsingular diagonal matrix \[ \mathbf{A} = \begin{pmatrix} a_{11} & 0 & 0 \ 0 & a_{22} & 0 \ 0 & 0 & a_{33} \end{pmatrix} \]The inverse \(\mathbf{A}^{-1}\) will be:\[ \mathbf{A}^{-1} = \begin{pmatrix} 1/a_{11} & 0 & 0 \ 0 & 1/a_{22} & 0 \ 0 & 0 & 1/a_{33} \end{pmatrix}. \]It’s crucial to ensure each diagonal element \(a_{ii} eq 0\) to avoid problems of division by zero, which would make the matrix singular, thereby preventing inversion. This principle holds for any \(n \times n\) diagonal matrix as well. Reciprocal adjustments of non-zero elements suffice to generate the inverse.

Determinant of a Matrix

The determinant is a special number that can be calculated from a square matrix. For diagonal matrices, this number holds a simple yet crucial role as it’s the product of all the diagonal elements. The determinant offers key insights about a matrix:- If it’s non-zero, the matrix is nonsingular and invertible.- If zero, the matrix is singular and lacks an inverse.For a simplified demonstration, let’s calculate the determinant of a \(3 \times 3\) diagonal matrix \[ \mathbf{A} = \begin{pmatrix} a_{11} & 0 & 0 \ 0 & a_{22} & 0 \ 0 & 0 & a_{33} \end{pmatrix}. \]The determinant is given by \[ det(\mathbf{A}) = a_{11} \cdot a_{22} \cdot a_{33}. \]For an \(n \times n\) diagonal matrix, the determinant is expanded as the product of all \(a_{ii}\) elements. Every non-zero \(a_{ii}\) ensures not only the matrix is nonsingular but guarantees the existence of its inverse as well.