Determinant Of Matrix C: Is It Invertible?

In this article, we'll walk through the process of evaluating the determinant of a given matrix C and then use that determinant to determine whether the matrix is invertible. Matrix invertibility is a fundamental concept in linear algebra, with wide-ranging applications in various fields such as computer graphics, cryptography, and economics. A matrix is invertible if and only if its determinant is non-zero. Let's dive in!

Understanding Matrix Determinants

Matrix determinants are scalar values that can be computed from square matrices. They provide crucial information about the matrix, including whether the matrix has an inverse (i.e., is invertible) and the volume scaling factor of the linear transformation described by the matrix. For a 3x3 matrix, the determinant can be calculated using several methods, such as cofactor expansion or row reduction. Let's focus on the cofactor expansion method, as it’s a widely used and straightforward approach. The determinant's value tells us a lot about the matrix's properties. A non-zero determinant means the matrix is invertible, while a zero determinant implies the matrix is singular (non-invertible).

To grasp the significance of matrix determinants, consider their role in solving systems of linear equations. A non-zero determinant ensures a unique solution exists for the system, making it invaluable in fields like engineering and economics where solving such systems is commonplace. Moreover, determinants are essential in eigenvalue problems, which are crucial in understanding the stability of systems in physics and control theory. The determinant essentially encapsulates the scaling factor of the transformation the matrix represents; a large determinant indicates significant scaling, while a small determinant suggests the transformation collapses space.

Determinants also help in change of variables in multivariable calculus, where they appear in the Jacobian determinant, facilitating the transformation of integrals from one coordinate system to another. In linear transformations, the determinant's sign indicates whether the transformation preserves the orientation of space (positive determinant) or reverses it (negative determinant). Understanding these properties of determinants is crucial not only for solving mathematical problems but also for applying these concepts in real-world scenarios. In data analysis, the determinant of the covariance matrix is used to measure the data's spread, and a low determinant might indicate multicollinearity issues. Thus, the determinant is a powerful tool with broad applicability.

Given Matrix C: An Overview

Before calculating the determinant, let's clearly define the matrix we're working with. We are given the matrix C as follows:

$$C=\begin{bmatrix} 1 & -3 & 17 \\ 1 & -1 & 7 \\ 2 & -5 & 29 \end{bmatrix}$$

This is a 3x3 matrix, which means it's a square matrix, and we can indeed calculate its determinant. Square matrices are fundamental in linear algebra because they represent linear transformations from a vector space to itself. The elements within the matrix are crucial as they define the specifics of this transformation. In the given matrix C, we have integer entries, which simplifies the arithmetic involved in calculating the determinant. Each row and column of the matrix represents a vector, and the determinant will give us insight into the linear independence of these vectors.

The entries of matrix C determine how it transforms vectors in three-dimensional space. For example, the first column ${[1, 1, 2]^T}$ tells us how the first basis vector ${[1, 0, 0]^T}$ is transformed, and similarly, the other columns correspond to the transformations of the second and third basis vectors. The determinant of C will tell us how the volume of a unit cube in the original space is scaled by this transformation. A positive determinant indicates that the orientation of space is preserved, while a negative determinant means the orientation is reversed. Zero determinant would imply that the matrix collapses the space, reducing its dimension, which means the columns are linearly dependent, and the matrix is not invertible.

Therefore, understanding the structure and entries of matrix C is the first step toward unraveling its properties through determinant calculation. This foundation allows us to predict the kind of transformation C enacts and whether this transformation can be reversed, which is essentially what invertibility means. The subsequent steps will involve specific computations, but this initial overview provides the context for why those computations are important.

Calculating the Determinant of Matrix C

To calculate the determinant of matrix C, we'll use the cofactor expansion method along the first row. The formula for the determinant of a 3x3 matrix using cofactor expansion is:

$$det(C) = a_{11}(C_{11}) - a_{12}(C_{12}) + a_{13}(C_{13})$$

Where a_ij represents the element in the i-th row and j-th column, and C_ij represents the cofactor of that element. The cofactor C_ij is calculated as (-1)^i+j times the determinant of the 2x2 matrix obtained by removing the i-th row and j-th column from the original matrix.

Let's apply this formula to our matrix C. The first element a₁₁ is 1. To find C₁₁, we remove the first row and first column, leaving us with the 2x2 matrix:

$$\begin{bmatrix} -1 & 7 \\ -5 & 29 \end{bmatrix}$$

The determinant of this 2x2 matrix is (-1 * 29) - (7 * -5) = -29 + 35 = 6. So, C₁₁ = 6.

Next, a₁₂ is -3. To find C₁₂, we remove the first row and second column, resulting in the 2x2 matrix:

$$\begin{bmatrix} 1 & 7 \\ 2 & 29 \end{bmatrix}$$

The determinant of this 2x2 matrix is (1 * 29) - (7 * 2) = 29 - 14 = 15. So, C₁₂ = (-1)¹⁺² * 15 = -15.

Finally, a₁₃ is 17. To find C₁₃, we remove the first row and third column, giving us the 2x2 matrix:

$$\begin{bmatrix} 1 & -1 \\ 2 & -5 \end{bmatrix}$$

The determinant of this 2x2 matrix is (1 * -5) - (-1 * 2) = -5 + 2 = -3. So, C₁₃ = -3.

Now, we substitute these values back into the determinant formula:

$$det(C) = 1 * 6 - (-3 * -15) + 17 * -3 = 6 - 45 - 51 = -90$$

Thus, the determinant of matrix C is -90. This non-zero determinant is a key piece of information for determining whether matrix C is invertible.

Determining Invertibility

Now that we have calculated the determinant of matrix C to be -90, we can determine whether the matrix is invertible. A matrix is invertible if and only if its determinant is non-zero. This is a fundamental theorem in linear algebra, linking the determinant, a scalar value, to a crucial property of the matrix.

In our case, the determinant of matrix C is -90, which is clearly not zero. Therefore, according to the theorem, matrix C is invertible. An invertible matrix, also known as a non-singular matrix, has several important properties. First and foremost, it means that there exists another matrix, called the inverse of C (denoted as C^-1), such that when C is multiplied by C^-1, the result is the identity matrix. This inverse matrix is unique and allows us to