Problem 13 - Differential Equations and Linear Algebra

Question

Use the cofactor expansion theorem to evaluate the given determinant along the specified row or column. \(\left|\begin{array}{rrrr}1 & -2 & 3 & 0 \\ 4 & 0 & 7 & -2 \\ 0 & 1 & 3 & 4 \\\ 1 & 5 & -2 & 0\end{array}\right|,\) column 4

Step-by-Step Solution

Answer

To evaluate the given determinant along column 4, find the cofactors by multiplying the minors of each element by their corresponding signs, then sum the products. The second element has a minor of -40, giving a cofactor of +80. The third element has a minor of -35, giving a cofactor of -140. The other elements in column 4 are zeros, hence their cofactors do not affect the resulting determinant. Summing the cofactors, we get \(-60\), and therefore, the determinant of the matrix is \(-60\).

Step 1: Identify the elements in the specified column

We are given a 4x4 matrix and asked to expand along column 4, so we need to identify the elements in column 4. In this case, they are 0, -2, 4, and 0.

Step 2: Find the minors of each element

Next, we need to find the 3x3 minors for each element in column 4. Since the first and last elements are zeros, their minors do not affect the resulting determinant. We will only need to find the minors for the second and third elements. For the second element, -2: \(|M_{24}| = \left|\begin{array}{rrr} 1 & -2 & 3 \\\ 0 & 1 & 3 \\\ 1 & 5 & -2\end{array}\right|\) For the third element, 4: \(|M_{34}| = \left|\begin{array}{rrr} 1 & -2 & 3 \\\ 4 & 0 & 7 \\\ 1 & 5 & -2\end{array}\right|\)

Step 3: Compute the cofactors

Now, compute the cofactors by multiplying the minors by their corresponding signs. The signs in column 4 are positive, negative, positive, negative. For the second element, -2: \(C_{24} = -(-2)(|M_{24}|)\) For the third element, 4: \(C_{34} = +(4)(|M_{34}|)\)

Step 4: Evaluate the minors

Next, evaluate the 3x3 determinants for the minors: \(|M_{24}| = 1(1(-2) - 3(5)) - (-2)(0(5) - 1(3)) + 3(0(1) - 4(3)) = -40\) \(|M_{34}| = 1(0(-2) - 7(5)) - (-2)(4(5) - 7(1)) + 3(4(1) - 0(1)) = -35\)

Step 5: Calculate the cofactors

To calculate the cofactors, plug the minors into the cofactor equations determined in Step 3: \(C_{24} = -(-2)(-40) = +80\) \(C_{34} = +(4)(-35) = -140\)

Step 6: Calculate the determinant using the cofactor expansion

Finally, sum the cofactors to get the determinant: \(|A| = 0 + 80 - 140 + 0 = -60\) The determinant of the given matrix is -60.

Key Concepts

Determinant

The determinant is a special number that can be computed from a square matrix. It is denoted as \(|A|\) for a matrix \(A\). Determinants are pivotal in matrix algebra as they help in finding the inverse of matrices, and in solving systems of linear equations. To calculate a determinant of a 4x4 matrix using cofactor expansion, choose a row or column, and proceed by breaking down the matrix into smaller components. The cofactor expansion is efficient for this purpose as it utilizes smaller 3x3 determinants, known as minors, along with the corresponding algebraic signs for evaluation.

Cofactor

A cofactor is a component used in the calculation of a determinant via the cofactor expansion method. It is derived from the minors of a matrix. To find a cofactor of an element in a matrix, you multiply the minor of that element by a sign, which depends on the element's position. The formula for finding the sign is \((-1)^{i+j}\), where \(i\) and \(j\) are the row and column numbers of the element, respectively. By calculating cofactors, you can reduce a complex determinant calculation into manageable steps, simplifying the arithmetic involved.

Minor (Matrix)

The minor of a matrix element is the determinant of the submatrix that remains after removing the row and column of that particular element. When evaluating a determinant by cofactor expansion, calculating the minor is an essential step. Take, for instance, the matrix in the exercise and the calculation of its minors: for each element in the chosen column, the corresponding submatrix is formed by excluding the row and column of that element, leading to a smaller 3x3 matrix from which the minor is determined. This process is crucial to streamline the complex computation of a larger matrix.

Matrix Algebra

Matrix algebra involves the manipulation and calculation of matrices. It is a crucial field of mathematics, especially in solving linear equations and transformations. Using concepts such as determinants and cofactors, matrix algebra provides the tools to understand and solve linear systems. When performing operations like the cofactor expansion, the knowledge of matrix algebra allows for a structured approach to breaking down a matrix into more understandable parts. This systematic decomposition aids in computing various attributes like determinants and ultimately enhances capability in tackling more complex mathematical problems.