Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

State conditions on \(a\) and \(b\) that guarantee that the matrix \(\left[\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right]\) has an inverse, and find a formula for the inverse if it exists.

Step-by-Step Solution

Answer

The conditions are \(a \neq 0\) and \(b \neq 0\); the inverse is \(\begin{bmatrix} \frac{1}{a} & 0 \\ 0 & \frac{1}{b} \end{bmatrix}\)."

Step 1: Identify Conditions for Matrix to be Invertible

For a 2x2 diagonal matrix \(\begin{bmatrix} a & 0 \ 0 & b \end{bmatrix}\) to have an inverse, both diagonal elements must be nonzero. This ensures that the determinant of the matrix is nonzero. Specifically, we need \( a eq 0 \) and \( b eq 0 \).

Step 2: Calculate the Determinant

The determinant of a diagonal matrix \(\begin{bmatrix} a & 0 \ 0 & b \end{bmatrix}\) is calculated as \( \text{det}(A) = a \times b \). The matrix is invertible if and only if the determinant is nonzero. In this case, \( a \cdot b eq 0 \).

Step 3: Write the Formula for the Inverse

If the matrix is invertible, its inverse can be found by taking the reciprocal of each nonzero diagonal element. Thus, the inverse of the matrix \(\begin{bmatrix} a & 0 \ 0 & b \end{bmatrix}\) is \(\begin{bmatrix} \frac{1}{a} & 0 \ 0 & \frac{1}{b} \end{bmatrix}\).

Key Concepts

Diagonal Matrix

A diagonal matrix is a type of square matrix where all the elements outside the main diagonal are zero. In formal terms, a matrix is diagonal if it is of the form:

• For a 2x2 matrix: \[\begin{bmatrix}a & 0 \0 & b\end{bmatrix}\]
• Here, \(a\) and \(b\) are the diagonal elements.

Diagonal matrices are among the simplest types of matrices in linear algebra. What makes them special is the ease with which we can perform operations like finding their determinants and inverses. Due to its structure, checking elements outside the diagonal helps immediately identify that they have a zero value, which can simplify many calculations involving matrices.

Determinant

The determinant is a special number calculated from a matrix, giving us essential information. For diagonal matrices, finding the determinant is particularly straightforward. The determinant of a 2x2 diagonal matrix, like the one discussed, is simply the product of its diagonal elements:

• For \[\begin{bmatrix}a & 0 \0 & b\end{bmatrix}\], the determinant is \( ext{det}(A) = a \times b \).

This result is crucial because a non-zero determinant indicates that the matrix is invertible. In our context, this means that both \(a\) and \(b\) must be non-zero for the matrix to have an inverse. Once you calculate this, you can quickly assess the matrix's invertibility status, making it an essential tool in matrix algebra.

Invertible Matrix

An invertible matrix, also known as a non-singular or non-degenerate matrix, is one that has an inverse. The existence of an inverse matrix is tied directly to the determinant of the matrix.For a diagonal matrix:

• The necessary and sufficient condition for a matrix to be invertible is that its determinant is not zero.
• This means each diagonal element must be non-zero. Specifically for our example: \(a eq 0\) and \(b eq 0\).

When these conditions are met, the matrix is invertible, meaning there exists another matrix that when multiplied with the original matrix yields the identity matrix. This is useful in solving linear equations and more complex matrix operations.

Inverse Matrix Formula

Finding the inverse of a diagonal matrix is simpler than other matrices because of its structure. The inverse of a matrix is a matrix such that when multiplied with the original, results in the identity matrix.For a diagonal matrix:

• Given the matrix \[\begin{bmatrix}a & 0 \0 & b\end{bmatrix}\], its inverse is \[\begin{bmatrix}\frac{1}{a} & 0 \0 & \frac{1}{b}\end{bmatrix}\].
• This computation involves taking the reciprocal of each non-zero diagonal element.

As both \(a\) and \(b\) are non-zero, you can easily compute their reciprocals to obtain the inverse. Inverting a diagonal matrix is straightforward and efficient, making it a preferred type of matrix for many applications in mathematics and engineering.