Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 6 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse is \(\begin{bmatrix} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{6} \end{bmatrix}\).

Step 1: Identify the Matrix Type

The matrix given is a diagonal matrix because all its non-diagonal elements (elements not on the main diagonal from top left to bottom right) are zero.

Step 2: Check for Invertibility

A square matrix is invertible if its determinant is non-zero. Since this is a diagonal matrix, the determinant can be calculated by multiplying all the diagonal elements: \(2 \times 4 \times 6 = 48\). Since the determinant is not zero, the matrix is invertible.

Step 3: Calculate the Inverse of the Diagonal Matrix

The inverse of a diagonal matrix can be easily calculated by taking the reciprocal of each of its diagonal elements, while keeping the non-diagonal elements as zero. Thus, the inverse matrix is \[\begin{array}{ccc}\frac{1}{2} & 0 & 0 \0 & \frac{1}{4} & 0 \0 & 0 & \frac{1}{6}\end{array}\]

Key Concepts

Diagonal Matrix

A diagonal matrix is a special type of square matrix where all the elements outside the main diagonal are zero. The main diagonal runs from the top left to the bottom right of the matrix. In simple terms, only a few numbers in specific positions have values, all others are zero. This makes diagonal matrices particularly easy to work with.

• Example: Consider the matrix \[\left[ \begin{array}{ccc}2 & 0 & 0 \0 & 4 & 0 \0 & 0 & 6 \end{array} \right]\]Here, 2, 4, and 6 lie on the main diagonal, and all other elements are zero.
• Attributes: Diagonal matrices are easy to identify because of their zero-filled structure outside the main diagonal.

Determinant Calculation

The determinant of a matrix is a special number calculated from its elements. It's especially straightforward for diagonal matrices. For any square matrix, if it has a non-zero determinant, it is invertible.

• In diagonal matrices, the determinant is computed by simply multiplying all the elements on the main diagonal. For example, in the matrix \[\left[ \begin{array}{ccc}2 & 0 & 0 \0 & 4 & 0 \0 & 0 & 6 \end{array} \right]\] the determinant is calculated as\[2 \times 4 \times 6 = 48.\]
• Invertibility Check: If the resultant determinant is not zero, like the 48 obtained above, the matrix is invertible.

Matrix Invertibility

A matrix is considered invertible if there exists another matrix, called the inverse, such that when multiplied together, they yield the identity matrix. For a matrix to be invertible, its determinant must be non-zero.

• Identify Non-Invertible Cases: A determinant of zero indicates that the matrix does not have an inverse, thus is singular.
• For our example matrix, the determinant was 48. This confirms that it is indeed invertible.

Matrix invertibility is a key concept for solving systems of linear equations, among other applications.

Inverse Calculation

Finding an inverse of a matrix means determining another matrix that can reverse the effect of the original when both matrices are multiplied together. With diagonal matrices, this is particularly simple.

• Calculate the Inverse: For diagonal matrices, take the reciprocal of each diagonal element, while all non-diagonal elements remain zero. For the above example matrix, the inverses of the elements are \[\left[ \begin{array}{ccc}\frac{1}{2} & 0 & 0 \0 & \frac{1}{4} & 0 \0 & 0 & \frac{1}{6} \end{array} \right]\]
• Verification: Upon multiplying these two matrices, you will obtain the identity matrix, confirming the correctness of the inverse calculation.

Calculating inverses is pivotal in various mathematical and engineering fields, streamlining processes that rely on reversible transformations.