Problem 65 - A Graphical Approach to Precalculus with Limits

Question

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

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

Step 1: Understanding the Matrix Type

The given matrix \( A \) is a diagonal matrix. A diagonal matrix is one where all off-diagonal elements are zero. Since \( A \) is diagonal and all diagonal elements are non-zero, \( A \) is invertible.

Step 2: Recognize Properties of Inverse for Diagonal Matrices

For a diagonal matrix of the form \( \left[\begin{array}{ccc}a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c\end{array}\right] \), its inverse is another diagonal matrix with each diagonal element being the reciprocal of the original, provided the original elements are not zero.

Step 3: Applying the Inverse Formula

Using the inverse property of diagonal matrices, compute \( A^{-1} \) by taking the reciprocals of the diagonal elements of \( A \). Thus, \( A^{-1} = \left[\begin{array}{ccc} \frac{1}{a} & 0 & 0 \ 0 & \frac{1}{b} & 0 \ 0 & 0 & \frac{1}{c} \end{array}\right] \).

Step 4: Verify the Inversion Condition

Multiply \( A \) by \( A^{-1} \) to confirm that it results in the identity matrix \( I \). Calculate \( A \times A^{-1} = \left[\begin{array}{ccc}a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{array}\right] \times \left[\begin{array}{ccc} \frac{1}{a} & 0 & 0 \ 0 & \frac{1}{b} & 0 \ 0 & 0 & \frac{1}{c} \end{array}\right] = \left[\begin{array}{ccc}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right] \). This confirms that the calculation of the inverse is correct.

Key Concepts

Matrix Inversion

Matrix inversion is the process of finding the matrix that, when multiplied with the original matrix, results in the identity matrix. This inverse matrix essentially "undoes" the effect of the original matrix. It's like finding the reciprocal of a number but in matrix terms. For a matrix to have an inverse, it must be "invertible" or "nonsingular," meaning all of its entries have a relationship where they can balance each other out as per matrix operations. If any essential condition isn't met, like determinant equals zero, the matrix can't be inverted.
Understanding matrix inversion is essential, especially in solving systems of linear equations and transformations in geometry.

• If a matrix is denoted as \( A \), then its inverse, when existent, is written as \( A^{-1} \).
• The multiplication of a matrix and its inverse, \( A \times A^{-1} \), should result in the identity matrix \( I \).

Therefore, knowing what makes a matrix invertible helps in activities that require reverting to original states, especially in computational mathematics and engineering.

Diagonal Matrix

A diagonal matrix is a special type of matrix where all the elements outside the main diagonal are zero. This structure simplifies many matrix operations, including addition and multiplication. Due to their format, diagonal matrices have unique properties, making them simpler to work with compared to full matrices.
One of the most exciting features of diagonal matrices is the straightforward calculation of their powers and inverses. Since calculations involve only the diagonal elements, they are computationally efficient.

• For a diagonal matrix \( A \) of order \( n \), only the elements \( a_{ii} \) where \( i = 1, 2, ..., n \) are nonzero.
• When it comes to finding the inverse, if the diagonal elements are non-zero, the inverse is simply calculated by taking the reciprocal of each non-zero diagonal element.

This property significantly simplifies the task of inverting diagonal matrices, as illustrated in the exercise solution.

Identity Matrix

The identity matrix is like the number 1 in matrix algebra. When any matrix is multiplied by an identity matrix, it remains unchanged. This matrix serves as an essential check for verifying the correctness of matrix inversions. An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere, representing a unit effect in multiplication.
In simpler terms, it acts as a multiplicative "do-nothing" in the world of matrices, just as multiplying a number by one does not change its value. For example, multiplying by identity matrix does not alter the original matrix:

• For a \( 3 \times 3 \) identity matrix \( I \), it is represented as:
  \[ I = \left[\begin{array}{ccc}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right] \]
• The identity matrix is crucial when verifying matrix inverses, as \( A \times A^{-1} = I \).

In the context of the provided exercise, the identity matrix verifies the correctness of the inverse by confirming that the product of \( A \) and \( A^{-1} \) yields \( I \).