Problem 5 - College Algebra with Corequisite Support

Question

Can a matrix with zeros on the diagonal have an inverse? If so, find an example. If not, prove why not. For simplicity, assume a \(2 \times 2\) matrix.

Step-by-Step Solution

Answer

Yes, a matrix with zeros on the diagonal can have an inverse; for example, \( A = \begin{bmatrix} 0 & 1 \\ 2 & 0 \end{bmatrix} \).

Step 1: Define a 2x2 matrix with zeros on the diagonal

A matrix with zeros on the diagonal takes the form: \( A = \begin{bmatrix} 0 & b \ c & 0 \end{bmatrix} \). Here, \( b \) and \( c \) are non-zero elements.

Step 2: Calculate the determinant

The determinant of matrix \( A \) is calculated as \( \text{det}(A) = (0)(0) - (b)(c) = -bc \). The determinant must be non-zero for the matrix to have an inverse.

Step 3: Check for a non-zero determinant

Since \( bc eq 0 \) for any non-zero \( b \) and \( c \), \( -bc eq 0 \). Thus, the determinant is non-zero, which implies that the matrix can have an inverse as long as both \( b \) and \( c \) are non-zero.

Step 4: Example of an invertible matrix with zeros on the diagonal

Consider the matrix \( A = \begin{bmatrix} 0 & 1 \ 2 & 0 \end{bmatrix} \). Its determinant is \( -2 \), which is non-zero. Therefore, this matrix is invertible.

Key Concepts

Diagonal Matrix

A diagonal matrix is one with non-zero elements only along its primary diagonal. This means all the off-diagonal elements are zero. In a general form, a diagonal matrix can be written as: \[D = \begin{bmatrix} d_1 & 0 & \ldots & 0 \ 0 & d_2 & & 0 \ \vdots & & \ddots & \vdots \ 0 & 0 & \ldots & d_n \end{bmatrix} \]Understanding diagonal matrices is crucial because they simplify many matrix-related calculations, such as determinants and inverses.

• They are particularly easy to work with since multiplying a vector by a diagonal matrix is straightforward and computationally efficient.
• Diagonal matrices are often used in diagonalization, which is the process of finding a similar diagonal matrix to a given matrix.

For the matrix \( A = \begin{bmatrix} 0 & b \ c & 0 \end{bmatrix} \), the zero positions on the diagonal are crucial as they impact the matrix's properties, like its determinant and inverse possibilities.

Determinant Calculation

The determinant of a matrix provides important insights into its properties. It is a special number that can tell us if a matrix is invertible and helps in computing the inverse. Determinants are vital in many areas of linear algebra, including systems of linear equations and eigenvalues. To find the determinant of a \(2 \times 2\) matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), you can use the formula:\[\text{det}(A) = ad - bc\]In our specific case, the matrix is \( A = \begin{bmatrix} 0 & b \ c & 0 \end{bmatrix} \), so the determinant becomes \(-bc\).

• The determinant must be non-zero for the matrix to have an inverse.
• A non-zero determinant indicates that the system of equations related to the matrix has a unique solution.

For matrix \( A \), as long as \( b \) and \( c \) are non-zero, \(-bc\) is non-zero, ensuring invertibility.

Matrix Inverse Condition

To find whether a matrix has an inverse, one must check two main conditions: the determinant must be non-zero, and the matrix must be square (having the same number of rows and columns).

• If the determinant of a matrix is zero, it is singular and doesn't have an inverse.
• A matrix inversion essentially "reverses" the matrix's operation, meaning multiplying a matrix by its inverse gives the identity matrix.

For the matrix \( A = \begin{bmatrix} 0 & b \ c & 0 \end{bmatrix} \), checking for non-zero determinant is key. With \( \text{det}(A) = -bc \), the matrix can have an inverse if both \( b \) and \( c \) are non-zero, such as in the example \( A = \begin{bmatrix} 0 & 1 \ 2 & 0 \end{bmatrix} \) with determinant \(-2\), allowing it to be invertible. Understanding how and when a matrix can be inverted is critical for solving equations and analyzing systems efficiently.