Question

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrr} 3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 4 \end{array}\right] $$

Short answer

The inverse of the given matrix is \[ \left[\begin{array}{rrr} \frac{1}{3} & 0 & 0 \ 0 & -\frac{1}{2} & 0 \ 0 & 0 & \frac{1}{4} \end{array}\right] \]

Step-by-step solution

Determine if the inverse exists

Check this by calculating the determinant of the matrix. For a diagonal matrix, the determinant is the product of the diagonal elements. In this case, it is \(3*-2*4 = -24\) which is different from zero, hence, the matrix is non-singular and its inverse exists.

Calculate the inverse of the matrix

For a diagonal matrix, the inverse is simply formed by taking the reciprocals of the diagonal elements. So, the inverse of the given matrix is obtained by replacing each diagonal entry \(a_{ii}\) with \(\frac{1}{a_{ii}}\). Therefore, the inverse matrix is as follows: \[ \left[\begin{array}{rrr} \frac{1}{3} & 0 & 0 \ 0 & -\frac{1}{2} & 0 \ 0 & 0 & \frac{1}{4} \end{array}\right] \]

Key concepts

Determinant of a Matrix

Understanding the determinant of a matrix is essential when dealing with matrix operations, especially when it comes to finding the inverse of a matrix. Essentially, the determinant is a scalar value that provides important mathematical properties of the matrix. For a square matrix, the determinant can help determine if the matrix is invertible.

In mathematical terms, the determinant of a matrix is calculated using a specific formula that depends on the size of the matrix. For a 2x2 matrix, the determinant is found by subtracting the product of its diagonals. For larger matrices, the calculation becomes more complex, involving permutations of matrix elements or employing techniques like Laplace expansion.

For diagonal matrices, the determinant simplifies to the product of its diagonal elements. This is because all the off-diagonal elements are zero, and thus their contribution to the determinant calculation through permutations is null. The fact that our exercise involves a diagonal matrix makes finding the determinant straightforward, and in our case, it results in -24. Since this value is non-zero, it indicates that the matrix indeed has an inverse, leading us to explore the next concept: non-singular matrices.

Non-Singular Matrix

The concept of a non-singular matrix is fundamental in linear algebra. A non-singular, or invertible, matrix is one that has an inverse. The inverse of a matrix, when multiplied by the original matrix, yields the identity matrix. The existence of an inverse matrix is dependent on the determinant of the original matrix; if the determinant is not zero, the matrix is non-singular.

When dealing with non-singular matrices, there are numerous methods to find the inverse, including algebraic techniques and computational algorithms. However, the exercise we are considering simplifies this process because it involves a diagonal matrix. In the context of the problem, confirming that the matrix is non-singular by checking that the determinant is -24 immediately tells us that we can proceed to find its inverse.

It's important to note that if we had found the determinant to be zero, the matrix would be considered singular, meaning no inverse exists. This would make the matrix incapable of certain operations and imply certain system equations have no unique solutions, an aspect crucial to understanding systems of linear equations.

Diagonal Matrix

A diagonal matrix is a specialized form of square matrix where all the elements outside the main diagonal are zero. This property greatly simplifies many matrix operations. Diagonal matrices are somewhat easier to analyze and work with, as many matrix properties can be readily inferred from the values of the diagonal elements alone.

In the context of finding the inverse, a diagonal matrix has a direct method. The inverse of a diagonal matrix is another diagonal matrix where each diagonal entry is the reciprocal of the original diagonal entry, provided none of these entries are zero (which would make the matrix singular). Since each off-diagonal element is already zero, and reciprocals do not affect zero values, they remain zero in the inverse matrix.

For our exercise, this means that each of the diagonal entries \(a_{ii}\) in the original matrix will simply be replaced by \(1/a_{ii}\) in the inverse matrix. This is a much simpler process compared to finding the inverse of non-diagonal matrices, showcasing one of the convenient aspects of diagonal matrices in linear algebra.