Question

Find the determinant of the given matrix using cofactor expansion along any row or column you choose. \(\left[\begin{array}{cccc}-1 & 0 & -2 & 5 \\ 3 & -5 & 1 & -2 \\ -5 & -2 & -1 & -3 \\ -1 & 0 & 0 & 0\end{array}\right]\)

Short answer

The determinant of the matrix is 13.

Step-by-step solution

Choose a Row or Column

Let's choose the last row for the cofactor expansion because it contains three zeros, making the calculation simpler. The matrix given is: \[ A = \begin{bmatrix} -1 & 0 & -2 & 5 \ 3 & -5 & 1 & -2 \ -5 & -2 & -1 & -3 \ -1 & 0 & 0 & 0 \end{bmatrix} \]

Identify Non-Zero Element in the Selected Row or Column

The only non-zero element in the last row is \(-1\). It is located in the first column of the last row (fourth row).

Find the Cofactor of the Non-Zero Element

We calculate the cofactor of the element \(-1\) in position (4,1). The minor matrix, after removing the 4th row and 1st column, is \[M_{41} = \begin{bmatrix}0 & -2 & 5 \ -5 & 1 & -2 \ -2 & -1 & -3 \end{bmatrix} \] The cofactor is \(C_{41} = (-1)^{4+1} \times ext{det}(M_{41}) = -1 \times ext{det}(M_{41})\)

Calculate the Determinant of the 3x3 Minor

To find \(\text{det}(M_{41})\), apply cofactor expansion along the first row of \[M_{41} = \begin{bmatrix}0 & -2 & 5 \ -5 & 1 & -2 \ -2 & -1 & -3 \end{bmatrix}\]Cofactor expansion gives: \(\text{det}(M_{41}) = 0 \times C_{11} - 2 \times C_{12} + 5 \times C_{13}\)Since the first term is zero, we have:\(= -2 \times \text{det}(\begin{bmatrix}-5 & -2 \ -2 & -3\end{bmatrix}) + 5 \times \text{det}(\begin{bmatrix}-5 & 1 \ -2 & -1\end{bmatrix})\)

Calculate the Det of 2x2 Minor M1 and M2

First, find the determinant of \(\begin{bmatrix}-5 & -2 \ -2 & -3\end{bmatrix}\): \(\begin{vmatrix} -5 & -2 \ -2 & -3 \end{vmatrix} = (-5)(-3) - (-2)(-2) = 15 - 4 = 11\)Now, calculate the determinant of \(\begin{bmatrix}-5 & 1 \ -2 & -1\end{bmatrix}\): \(\begin{vmatrix} -5 & 1 \ -2 & -1 \end{vmatrix} = (-5)(-1) - (1)(-2) = 5 + 2 = 7\)

Plug Values Back Into the Cofactor Expansion

Substitute the values obtained from the determinants of the minor matrices:\(\text{det}(M_{41}) = -2 \times 11 + 5 \times 7\)Calculate:\(= -22 + 35 = 13\)

Calculate the Cofactor and Final Determinant

Substitute the value of \(\text{det}(M_{41})\) into the cofactor formula:\(\text{C}_{41} = -1 \times 13 = -13\)Thus, the determinant of the entire 4x4 matrix \(A\) is\(\text{det}(A) = (-1) \times \text{C}_{41} = -1 \times -13 = 13\)

Conclusion

The determinant of the matrix is computed using cofactor expansion along the fourth row, resulting in a value of 13.

Key concepts

Cofactor Expansion

Cofactor expansion is a method used in matrix algebra to find the determinant of a matrix. It's a process where you select a specific row or column of a matrix and express the determinant as a sum of products. Cofactor expansion is particularly useful for larger matrices because it breaks down the determinant calculation into simpler parts.

• Choose a row or column from the matrix to simplify the calculation. Often, selecting a row or column with zeros minimizes computation.
• For each element in that row or column, calculate the cofactor. The cofactor is the signed determinant of the submatrix you get by removing the row and the column of the current element.
• The formula for the cofactor of an element at position (i, j) is: \[ C_{ij} = (-1)^{i+j} \times \text{det}(M_{ij}) \ \/where \( \text{det}(M_{ij}) \) is the determinant of the minor matrix. \]
• The determinant of the matrix is the sum of the products of each element in the chosen row or column and its corresponding cofactor.

This technique is also called the Laplace's expansion and, when used strategically, it reduces complex calculations dramatically, especially in sparse matrices.

Minor Matrix

The minor matrix is a key concept in understanding determinant calculations through cofactor expansion. A minor is derived from a larger matrix by eliminating one row and one column. Specifically, for any element located in a given row i and column j of matrix A, its minor matrix M_{ij} is formed by deleting row i and column j from matrix A.

• To form the minor of a matrix element, remove the row and column of that element.
• The resulting smaller matrix, typically (n-1)x(n-1) from an nxn matrix, contains all the elements that were not in the removed row and column.
• Example: For a 4x4 matrix, a minor matrix will be 3x3.

Calculating the determinant of a minor matrix is crucial because it directly affects the cofactor terms, which in turn, contribute to the original matrix's determinant. Thus, understanding how to accurately determine minors is a foundational skill in linear algebra.

Matrix Algebra

Matrix algebra involves a variety of operations and techniques applied to matrices, which are rectangular arrays of numbers, symbols, or expressions. Matrix algebra is fundamental in solving linear equations, transforming geometrical objects, and performing various data manipulations. It includes operations like matrix addition, subtraction, multiplication, and finding determinants.

• Addition: Matrices are added by adding their corresponding elements. Matrices must be of the same dimension.
• Multiplication: Involves dot products of rows and columns. Unlike addition, it requires the appropriate number of columns from the first matrix to match the number of rows in the second matrix.
• Transpose: Flips a matrix over its diagonal, exchanging rows with columns.
• Determinant: A scalar value that is a function of a square matrix, expressing certain properties of the matrix.

Matrix algebra is indispensable in higher mathematics and applied fields like physics and engineering for modeling complex systems and solving differential equations.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. It is the foundation of much of modern mathematics and many applications across science and engineering. Linear algebra involves operations on vectors and matrices, exploring concepts like vector spaces, linear transformations, and eigenvalues.

• Vector Spaces: Collections of vectors where vector addition and scalar multiplication are defined.
• Linear Transformations: Functions that map vectors to vectors in a manner that preserves vector addition and scalar multiplication.
• Eigenvalues and Eigenvectors: For a given transformation, eigenvectors maintain their direction while eigenvalues are scalars showing how the magnitude of the eigenvectors is changed.
• Systems of Linear Equations: Solved using various linear algebra techniques, crucial for numerous applications in computer science, economics, and more.

In essence, linear algebra provides the tools to model and solve high-dimensional systems, making it a critical component in data analysis and computational fields.