Question

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{array}\right] $$

Short answer

The inverse of the given matrix is \[\left[\begin{array}{ccc} 1/2 & 0 & 0 \ 0 & 1/3 & 0 \ 0 & 0 & 1/5 \end{array}\right] \]

Step-by-step solution

- Check if inverse exists

First, we need to check whether the inverse of the given matrix exists or not. A matrix will have an inverse if and only if its determinant is not equal to zero. The determinant of a diagonal matrix, like the one in this problem, is the product of its diagonal elements, i.e., \(2*3*5 = 30\). Since the determinant is not zero, the given matrix does have an inverse.

- Find the inverse of the matrix

To find the inverse of a diagonal matrix, we simply need to take the reciprocal of each of the diagonal elements. Remember to leave the non-diagonal elements as they are (i.e., zero). This leads us to the inverse matrix: \[\left[\begin{array}{ccc} 1/2 & 0 & 0 \ 0 & 1/3 & 0 \ 0 & 0 & 1/5 \end{array}\right] \]

- Verify the Inverse Matrix

To determine if this is indeed the inverse, we could multiply the original matrix by this supposed 'inverse'. The result should be the identity matrix. However, given that we are dealing with a diagonal matrix and we've checked that the determinant is not zero, we can be certain that we've found the inverse correctly.

Key concepts

Determinant of a Matrix

The determinant of a matrix is a special number which provides important information about the matrix. In the context of a square matrix, the determinant can tell us whether a matrix is invertible (i.e., whether an inverse exists).
For a diagonal matrix, the determinant is especially simple to calculate. It involves multiplying all the values down the main diagonal. For example, given the matrix in the exercise:

• The entries are 2, 3, and 5 on the diagonal.
• The determinant is calculated as \(2 \times 3 \times 5 = 30\).

A matrix with a determinant of zero cannot be inverted, meaning no inverse matrix exists. However, if the determinant is not zero (like in our exercise), an inverse matrix does exist.
Understanding the determinant is crucial as it directly influences matrix operations and potential solutions to matrix equations.

Diagonal Matrix

A diagonal matrix is a type of matrix where all non-diagonal elements are zero. This means:

• The only nonzero entries appear along the top-left to bottom-right diagonal line.
• These matrices are simple to work with due to their sparse nature.
• When computing operations such as the determinant or finding an inverse, these matrices offer a convenient advantage.

In our exercise, the matrix \(\left[\begin{array}{lll}2 & 0 & 0 \0 & 3 & 0 \0 & 0 & 5\end{array}\right]\) is diagonal. Operations like inversion become straightforward; just focus on the diagonal elements. You only switch the diagonal elements to their reciprocal when finding the inverse. Therefore, this specific characteristic makes many computations, like solving linear equations or finding eigenvalues, quite efficient.

Identity Matrix

The identity matrix plays a crucial role in matrix algebra, acting like the number 1 does in basic arithmetic. An identity matrix has ones on the diagonal and zeros elsewhere.
When you multiply any square matrix by an identity matrix of compatible size, it returns the original matrix unchanged. For example, if \(I\) is an identity matrix and \(A\) is any matrix of the same size, then:

• \(IA = A\)
• \(AI = A\)

In dealing with matrix inverses, a correct inverse multiplied by the original matrix yields an identity matrix. This confirms that the inverse was calculated correctly. In this exercise, the multiplication of the original diagonal matrix by its inverse should result in the identity matrix. Hence understanding the identity matrix ensures that verification processes are correctly performed after calculations like finding inverses.