Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the inverse of the matrix \(\left[\begin{array}{rr}2 & 0 \\ 3 & -1\end{array}\right],\) if it exists; otherwise, state that the matrix is singular.

Short answer

The inverse of the matrix is \(\left[\begin{array}{rr}\frac{1}{2} & 0 \ \frac{3}{2} & -1\end{array}\right].\)

Step-by-step solution

Calculate Determinant

First, determine if the matrix has an inverse by calculating the determinant. For matrix \(\left[\begin{array}{rr}2 & 0 \ 3 & -1\end{array}\right]\), the determinant is calculated as follows: \[\text{det} = (2 \cdot -1) - (0 \cdot 3) = -2.\]

Check if the Matrix is Singular

A matrix is singular if its determinant is 0. Since the determinant we calculated is -2, which is not 0, the matrix is not singular and an inverse exists.

Find the Adjugate Matrix

Next, find the adjugate of the matrix. The adjugate is constructed by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. For the matrix \(\left[\begin{array}{rr}2 & 0 \ 3 & -1\end{array}\right]\), the adjugate is: \[adj(A) = \left[\begin{array}{rr}-1 & 0 \ -3 & 2\end{array}\right].\]

Calculate the Inverse Matrix

The inverse of a matrix is given by the formula \(A^{-1} = \frac{1}{\text{det}} \cdot \text{adj}(A).\) Substitute the determinant and the adjugate matrix to get: \[A^{-1} = \frac{1}{-2} \cdot \left[\begin{array}{rr}-1 & 0 \ -3 & 2\end{array}\right] = \left[\begin{array}{rr}\frac{-1}{-2} & \frac{0}{-2} \ \frac{-3}{-2} & \frac{2}{-2}\end{array}\right] = \left[\begin{array}{rr}\frac{1}{2} & 0 \ \frac{3}{2} & -1\end{array}\right].\]

Key concepts

Determinant

The determinant of a matrix is a special number that can tell us if a matrix has an inverse or not. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated as \(\text{det} = ad - bc\).

For the matrix given in the exercise, \(\begin{bmatrix} 2 & 0 \ 3 & -1 \end{bmatrix}\), the determinant is calculated as follows:

• Multiply 2 by -1
• Multiply 0 by 3

So, the determinant is \(-2 - 0 = -2\).

The determinant helps us determine if the matrix is singular. If the determinant is zero, the matrix is singular and does not have an inverse. Since our determinant is -2, which is not zero, the matrix is not singular and an inverse exists.

Adjugate Matrix

To find the inverse of a matrix, we use the adjugate matrix. The adjugate, or adjoint, of a matrix is formed by swapping its main diagonal elements and changing the signs of its off-diagonal elements.

For the matrix \(\begin{bmatrix} 2 & 0 \ 3 & -1 \end{bmatrix}\), the adjugate is calculated as:

• Swap the elements on the main diagonal: place -1 where 2 was, and place 2 where -1 was.
• Change the signs of the off-diagonal elements: 0 remains 0 (as it's sign change doesn't affect it) and 3 becomes -3.

So, the adjugate matrix for \(\begin{bmatrix} 2 & 0 \ 3 & -1 \end{bmatrix}\) is: \(\begin{bmatrix} -1 & 0 \ -3 & 2 \end{bmatrix}\).

The adjugate matrix is a crucial step in finding the inverse of a matrix.

Singular Matrix

A matrix is singular if it does not have an inverse. This occurs when its determinant is zero.

From our exercise, the determinant of the matrix \(\begin{bmatrix} 2 & 0 \ 3 & -1 \end{bmatrix}\) is -2, which is not zero.

Thus, our matrix is not singular.

In general, singular matrices are important in matrix algebra because they cannot be inverted, which means they lose some very useful properties. Ensuring that a matrix is non-singular (i.e., its determinant is not zero) is key when solving systems of linear equations or performing other operations where the inverse is needed.

Matrix Algebra

Matrix algebra refers to the collection of operations that can be performed on matrices. These include addition, subtraction, multiplication (both by a scalar and matrix multiplication), finding determinants, adjugates, and inverses, among others.

Understanding matrix algebra is essential for solving linear equations, performing transformations in geometry, and in numerous applications in engineering, physics, and computer science.

Let's summarize the steps used in the exercise to find the inverse of a matrix using matrix algebra:

• Determine the determinant of the matrix.
• Check if the matrix is singular by examining the determinant.
• If the matrix is non-singular (determinant not zero), find the adjugate matrix.
• Compute the inverse by multiplying the adjugate by the reciprocal of the determinant.

Mastering these steps allows for deeper insights and solutions in various mathematical problems.