Problem 79 - Precalculus

Question

Evaluate the determinant: $$ \left|\begin{array}{rrrr} 1 & -2 & -1 & 3 \\ 4 & 0 & 1 & 2 \\ 0 & 3 & 2 & 4 \\ 1 & -3 & 5 & -4 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is -268.

Step 1: Select a Row or Column

The determinant can be simplified by expanding along a row or column with zeros or simpler numbers. The first row is a good choice here because it starts with 1, which will make calculations easier.

Step 2: Expand Along the First Row

Expanding along the first row, apply the formula for determinant expansion: \[ \det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} + a_{14}C_{14} \] For our matrix, the expansion along the first row is: \[ = 1\cdot C_{11} - 2\cdot C_{12} - 1\cdot C_{13} + 3\cdot C_{14} \] where \(C_{ij}\) is the cofactor of element \(a_{ij}\).

Step 3: Calculate Minors for Cofactors

Compute the minors by removing the row and column of each element and finding the determinant of the resulting 3x3 matrix: - \( M_{11} = \left| \begin{array}{rrr} 0 & 1 & 2 \ 3 & 2 & 4 \ -3 & 5 & -4 \end{array} \right| \)- \( M_{12} = \left| \begin{array}{rrr} 4 & 1 & 2 \ 0 & 2 & 4 \ 1 & 5 & -4 \end{array} \right| \)- \( M_{13} = \left| \begin{array}{rrr} 4 & 0 & 2 \ 0 & 3 & 4 \ 1 & -3 & -4 \end{array} \right| \)- \( M_{14} = \left| \begin{array}{rrr} 4 & 0 & 1 \ 0 & 3 & 2 \ 1 & -3 & 5 \end{array} \right| \)

Step 4: Calculate Determinants of 3x3 Matrices

Each 3x3 matrix must be solved using the formula for determinants of a 3x3 matrix:\[ \left| \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array} \right| = a(ei - fh) - b(di - fg) + c(dh - eg) \]Using this formula, we calculate:- \( M_{11} = (0)(2)(-4) + (3)(5)(2) + (-3)(-1)(2) = -46 \)- \( M_{12} = (4)(2)(-4) + (1)(4)(1) + (2)(0)(5) = -36 \)- \( M_{13} = (4)(3)(-4) + (0)(4)(1) + (2)(-3)(0) = -48 \)- \( M_{14} = (4)(3)(5) + (0)(2)(1) + (1)(0)(-3) = 66 \)

Step 5: Calculate Cofactors

The cofactor for each element is obtained by applying the sign as follows: - \( C_{11} = (-1)^{1+1}M_{11} = (-1)^2 M_{11} = -46 \)- \( C_{12} = (-1)^{1+2}M_{12} = (-1)^3 M_{12} = 36 \)- \( C_{13} = (-1)^{1+3}M_{13} = (-1)^4 M_{13} = -48 \)- \( C_{14} = (-1)^{1+4}M_{14} = (-1)^5 M_{14} = -66 \)

Step 6: Evaluate Determinant

Substitute the cofactors back into the expansion formula:\[ \det(A) = 1\cdot(-46) - 2\cdot(36) - 1\cdot(-48) + 3\cdot(-66) \]Simplifying, we have:\[ = -46 - 72 + 48 - 198 \]\[ = -268 \]

Key Concepts

Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a technique used to calculate the determinant of a square matrix. This method involves breaking down a matrix into simpler components, which makes it easier to handle complex matrices. Particularly, we select either a row or a column, and use the elements of that row or column combined with their cofactors to expand the determinant.

• Each element in the chosen row or column is multiplied by its respective cofactor.
• The cofactor of an element is defined as the determinant of the submatrix that remains after removing the element's row and column, multiplied by \((-1)^{i+j}\), where \i\ and \j\ are the row and column indices of the element.

Notice the alternating sign pattern applied to cofactors. This is due to the \((-1)^{i+j}\) factor, which affects the sign of the cofactor based on the position of the element within the matrix.

3x3 Matrix Determinant

Once a 4x4 matrix is expanded using cofactor expansion, you'll often end up calculating determinants of smaller, 3x3 matrices. Understanding how to find the determinant of these 3x3 matrices is crucial for solving the original problem.
The process involves:

• Consider a 3x3 matrix \( \left| \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array}\right| \) and use the formula: \(a(ei - fh) - b(di - fg) + c(dh - eg)\).
• This formula essentially breaks down a 3x3 matrix into three simpler components using the rule of Sarrus.

Sarrus's rule is a visual method that emphasizes taking diagonals of the matrix. It applies to 3x3 matrices and provides an efficient way to calculate determinants by considering both main diagonals and secondary diagonals.

Laplace Expansion

Laplace expansion, also referred to as cofactor expansion, is used for evaluating determinants of larger matrices by breaking them into smaller matrices. This is useful for matrices larger than 3x3, like 4x4 matrices, where direct computation isn't feasible.
Key points about Laplace expansion:

• The choice of which row or column to expand along can simplify calculations substantially, especially when that row or column contains zeros or simpler numbers.
• The sum of the products of each element of the row/column and its corresponding cofactor leads to the determinant of the matrix.

An efficient strategy is to pick the row or column with most zeros, as each zero simplifies the calculations by eliminating the need to compute one minor. Understanding this concept allows for the practical computation of determinants in various applications across science and engineering.

Precalculus

Precalculus is an essential foundation for understanding many mathematical concepts, including matrices and their determinants. Determinants are a part of linear algebra, which is extensively used in higher-level calculus and beyond.
These concepts form the building blocks of solving systems of equations, transforming geometric figures, and analyzing linear transformations. Some practical applications of matrix determinants include:

• Checking whether a system of linear equations has a unique solution.
• Determining the invertibility of a matrix, as a matrix is invertible only if its determinant is non-zero.

For students preparing for calculus, a strong grasp of precalculus topics like determinants and matrix algebra is critical. It will not only ease your transition into calculus but also help in fields that utilize advanced mathematics.