Problem 65 - A Graphical Approach to College Algebra

Question

Given a square matrix \(A^{-1}\), find matrix \(A\). Let \(A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right],\) where \(a, b,\) and \(c\) are nonzero real numbers. Find \(A^{-1}\).

Step-by-Step Solution

Answer

\(A^{-1} = \left[ \begin{array}{ccc} \frac{1}{a} & 0 & 0 \\ 0 & \frac{1}{b} & 0 \\ 0 & 0 & \frac{1}{c} \end{array} \right]\)."

Step 1: Understanding the Problem

We are given the expression for a square matrix \(A\) and need to find its inverse, \(A^{-1}\). We must ensure \(A\) is invertible, which it is because all diagonal elements \(a, b, c\) are nonzero. To find \(A^{-1}\), we apply the inverse formula for a diagonal matrix.

Step 2: Diagonal Matrix Inverse Formula

For any diagonal matrix \(D = \left[ \begin{array}{ccc} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{array} \right]\), the inverse \(D^{-1}\) is \(\left[ \begin{array}{ccc} \frac{1}{a} & 0 & 0 \ 0 & \frac{1}{b} & 0 \ 0 & 0 & \frac{1}{c} \end{array} \right]\), given that all diagonal elements are non-zero.

Step 3: Calculate A^{-1} Using the Formula

Applying the inverse formula to matrix \(A\), we calculate \(A^{-1} = \left[ \begin{array}{ccc} \frac{1}{a} & 0 & 0 \ 0 & \frac{1}{b} & 0 \ 0 & 0 & \frac{1}{c} \end{array} \right]\). This matrix is the inverse of \(A\).

Key Concepts

Diagonal Matrix

A diagonal matrix is a type of square matrix where all off-diagonal elements are zero. In simpler terms, this means that all the elements that are not on the main diagonal (the one that runs from the top left to the bottom right) are zero. Diagonal matrices are particularly easy to work with in linear algebra
because:

• Their determinant is the product of the diagonal elements.
• They are always invertible if none of the diagonal elements are zero.
• Their inverse consists of taking the reciprocal of each non-zero diagonal element.

In our example, the matrix \(A\), is diagonal with non-zero elements \(a, b, c\) on the main diagonal, making it both invertible and easy to manage mathematically.

Inverse Matrix Formula

The inverse of a matrix is a crucial aspect in linear algebra. It helps solve the equation \(AX = I\), where \(A\) is a matrix, \(X\) is unknown, and \(I\) is the identity matrix.
For diagonal matrices, such as matrix \(A\) given in the problem, there is a simple formula for finding the inverse. When \(D\) is a diagonal matrix with elements \(a, b, c\), the inverse \(D^{-1}\) is another diagonal matrix where each diagonal element is replaced by its reciprocal:
\[ D^{-1} = \begin{bmatrix} \frac{1}{a} & 0 & 0\ 0 & \frac{1}{b} & 0\ 0 & 0 & \frac{1}{c} \end{bmatrix} \]
This formula is only valid if none of the diagonal elements are zero, ensuring that the reciprocal is defined.

Square Matrix

A square matrix is a matrix that has the same number of rows and columns. This characteristic is important in various linear algebra applications, particularly when determining invertibility.
In our specific context, the square matrix \(A\) has dimensions 3x3 and is essential to consider when
finding its inverse:

• Only square matrices can have inverses. Non-square matrices are not invertible.
• To be invertible, a square matrix must have a non-zero determinant.

Square matrices also make it possible to discuss the concept of diagonal matrices, as diagonals only make sense in this configuration. This inherent structure simplifies calculations such as determinant and inverses, as demonstrated in the provided exercise.