Question

Evaluate determinant by calculator or by minors. $$\left|\begin{array}{rrrr}1 & 2 & -1 & 1 \\\\-1 & 1 & 2 & 3 \\\3 & -1 & 1 & 2 \\\1 & 2 & -1 & 1\end{array}\right|$$

Short answer

The determinant can be calculated by using a calculator with matrix functionality or, if necessary, by expanding along a row or column using the method of minors and cofactors.

Step-by-step solution

Calculate determinant directly using calculator

If a calculator is available that can handle matrix operations, input the matrix and use the determinant function to calculate the result.

Calculate determinant by minors

If a calculator is not available or for practice, expand the determinant using the method of minors. Choose a row or column, for example, the first column, and calculate its minors and cofactors. Then multiply each element of the column by its corresponding cofactor and take the sum to find the determinant.

Expand along the first column

Take the first element of the first column (1), multiply by the determinant of the minor matrix obtained by removing the first row and column. Then subtract, the second element of the first column (-1) multiplied by the determinant of the minor matrix obtained by removing the second row and column. Continue in this manner until the end of the column.

Write out the expansion

The expansion along the first column is \[ 1 \cdot \left|\begin{matrix}1 & 2 & 3\-1 & 1 & 2\2 & -1 & 1\end{matrix}\right| - (-1) \cdot \left|\begin{matrix}2 & -1 & 1\1 & 2 & 3\2 & -1 & 1\end{matrix}\right| + 3 \cdot \left|\begin{matrix}2 & -1 & 1\1 & 1 & 3\1 & 2 & 1\end{matrix}\right| - 1 \cdot \left|\begin{matrix}2 & -1 & 1\1 & 1 & 2\-1 & 1 & 2\end{matrix}\right| \]

Calculate each minor's determinant

The determinant of each 3x3 minor matrix can be found by applying the same method of minors or by using the rule of Sarrus, since they are smaller matrices.

Sum up the result

Sum up the results of each cofactor multiplied by its corresponding matrix element to get the determinant of the original 4x4 matrix.

Key concepts

Determinant by Minors

Calculating the determinant by minors is a fundamental procedure in linear algebra, essential for understanding the properties of a matrix. This method involves breaking down a larger matrix into smaller ones, which are referred to as 'minors'. To find a minor, you remove one row and one column from the original matrix. For the given 4x4 matrix, it's often helpful to start with the first column or row, as they may contain zeros or ones which simplify calculations.

The determinant of the minor, a smaller square matrix, is then calculated, an operation that is recursively applied if the minor is still larger than a 2x2 matrix. When minors are created from a 3x3 matrix, as in Step 5 of this exercise, they can be calculated using more straightforward methods such as the rule of Sarrus or by applying minors again if it feels more intuitive.

For improvement and deep understanding, it's advantageous to practice calculating the determinant of various minors by hand to master the process for different sizes of matrices, especially since this conceptual understanding aids in both theoretical and applied mathematics.

Matrix Operations

Matrix operations include a variety of computations, such as addition, subtraction, multiplication, and the calculation of a determinant, each carrying its own set of rules. Understanding how to manipulate matrices is crucial in linear algebra, as it forms the basis of more complex operations and applications in various fields, from physics to economics.

A key operation is finding the determinant of a matrix, which gives important information about the system it represents, such as whether solutions exist for a set of equations or if the matrix is invertible. The procedure for calculating a determinant varies depending on the size of the matrix. For large matrices, like the 4x4 matrix in our exercise, it often involves simplification techniques, such as row and column operations that can help reduce the matrix to a smaller size, making the determinant easier to compute.

In practice, the computational load for larger matrices can be significant, and thus software or a calculator often performs these operations. When learning matrix operations, it's beneficial to use technology as a tool to check manual computations and to gain a deeper understanding of the outcomes and patterns involved in these operations.

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a technique used to calculate the determinant of a matrix. It expands the determinant into a sum of products, where each product is the multiplication of an element from a row or column with its corresponding cofactor.

The cofactor itself is determined by taking the determinant of the minor associated with that matrix element and then applying a sign that depends on the element's position within the matrix, following a checkerboard pattern of positive and negative signs. This sign is critical since it ensures the terms are added and subtracted in the correct sequence, maintaining the integrity of the determinant's calculation.

In the case of the 4x4 matrix provided in the exercise, the expansion progresses along the first column. Each step multiplies the element from the first column with the determinant of the respective minor, while incorporating the sign change for cofactors. To facilitate the understanding of cofactor expansion, visual aids that illustrate the checkerboard pattern of signs or interactive tools that allow you to remove rows and columns can be very useful in grasping the concept.

While the algebraic process can be intricate for larger matrices, developing a solid grasp of cofactor expansion ensures a robust mathematical foundation that supports a variety of applications in higher mathematics, physics, and engineering.