Problem 33 - Precalculus: Mathematics for Calculus

Question

Use a graphing calcuIator to find the inverse of the matrix, if it exists. $$\left[\begin{array}{llll}1 & 0 & 0 & 0 \\\0 & 2 & 0 & 0 \\\0 & 0 & 4 & 0 \\\0 & 0 & 0 & 7\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse is \[\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0.5 & 0 & 0 \\ 0 & 0 & 0.25 & 0 \\ 0 & 0 & 0 & 0.142857 \end{bmatrix}\].

Step 1: Understanding Matrix and Inverse

We are given a 4x4 diagonal matrix: \[A = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 2 & 0 & 0 \ 0 & 0 & 4 & 0 \ 0 & 0 & 0 & 7 \end{bmatrix}\].For a matrix to have an inverse, it must be square (which it is) and have a non-zero determinant.

Step 2: Calculate Determinant of the Matrix

The determinant of a diagonal matrix is the product of its diagonal elements. Here, the determinant is \(det(A) = 1 \times 2 \times 4 \times 7\). Calculating this, we get \(det(A) = 56\). Since 56 is non-zero, the inverse exists.

Step 3: Find the Inverse of a Diagonal Matrix

The inverse of a diagonal matrix is simply another diagonal matrix with the reciprocal of each diagonal element from the original matrix. Thus, the inverse matrix \(A^{-1}\) is: \[A^{-1} = \begin{bmatrix} 1/1 & 0 & 0 & 0 \ 0 & 1/2 & 0 & 0 \ 0 & 0 & 1/4 & 0 \ 0 & 0 & 0 & 1/7 \end{bmatrix}\], which simplifies to \[ \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 0.5 & 0 & 0 \ 0 & 0 & 0.25 & 0 \ 0 & 0 & 0 & 0.142857 \end{bmatrix}\].

Key Concepts

Diagonal Matrix

A diagonal matrix is a special kind of matrix where most of the elements are zero. The non-zero elements are arranged along the diagonal from the top left to the bottom right. This type of matrix looks like a ladder, with non-zero steps and the rest being zeros.
Diagonal matrices are easy to work with because you can quickly identify important properties such as the determinant and the inverse, if it exists.

• All entries outside the diagonal are zero.
• Common in simplifying linear equations and transformations.
• Often used as building blocks in more complex matrix equations.

Determinant

The determinant is a special number calculated from a square matrix. It gives us critical information about the matrix, like whether it has an inverse.
For diagonal matrices, calculating the determinant is straightforward. Simply multiply all the diagonal elements together. For example, if your diagonal elements are 1, 2, 4, and 7, their product will give you the determinant.

• A zero determinant means the matrix has no inverse.
• It helps in understanding the properties and functionalities of matrices.
• In geometric terms, the determinant relates to changes in volume or area when a transformation is applied.

Inverse Matrix

The inverse of a matrix is similar to the reciprocal of a number. This matrix, when multiplied by the original matrix, gives the identity matrix—like how a number times its reciprocal equals one.
Finding the inverse of a diagonal matrix is quick and easy. You simply find the reciprocal of each non-zero diagonal element and form a new diagonal matrix.

• If a matrix has an inverse, it is called "invertible" or "non-singular."
• The inverse matrix is used in solving systems of equations.
• Not all matrices have an inverse; the determinant helps us identify which ones do.

Reciprocal

The term reciprocal generally refers to taking one divided by a number, resulting in its inverse. In the context of matrices, the reciprocal relates to the elements along the diagonal of a diagonal matrix.
Here, to find the inverse of a diagonal matrix, you calculate the reciprocal of each diagonal element. This means, in a matrix with diagonal numbers 1, 2, 4, and 7, you transform them to 1, 0.5, 0.25, and approximately 0.142857 to get the inverse matrix.

• Working with reciprocals can simplify complex matrix computations.
• The reciprocal operation is central to finding the inverse.
• A reciprocal of a number is essentially a multiplicative inverse.