Problem 28 - Algebra and Trigonometry

Question

Evaluate the determinant, using row or column operations whenever possible to simplify your work. $$ \left|\begin{array}{rrrr}{-2} & {3} & {-1} & {7} \\ {4} & {6} & {-2} & {3} \\\ {7} & {7} & {0} & {5} \\ {3} & {-12} & {4} & {0}\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the matrix is 604.

Step 1: Expand across the first row

To simplify calculating the determinant, let's expand along the first row. The first row of the matrix is \([-2, 3, -1, 7]\). We'll evaluate the determinant by calculating the cofactors for each element of the first row.

Step 2: Calculate cofactors

For each element in the first row, calculate the cofactor by eliminating the row and column that the element resides and find the determinant of the resulting 3x3 matrix.- Cofactor of \(-2\): Remove the first row and first column: \[\begin{vmatrix} 6 & -2 & 3 \ 7 & 0 & 5 \ -12 & 4 & 0 \end{vmatrix}\]- Cofactor of \(3\): Remove the first row and second column: \[\begin{vmatrix} 4 & -2 & 3 \ 7 & 0 & 5 \ 3 & 4 & 0 \end{vmatrix}\]- Cofactor of \(-1\): Remove the first row and third column: \[\begin{vmatrix} 4 & 6 & 3 \ 7 & 7 & 5 \ 3 & -12 & 0 \end{vmatrix}\]- Cofactor of \(7\): Remove the first row and fourth column: \[\begin{vmatrix} 4 & 6 & -2 \ 7 & 7 & 0 \ 3 & -12 & 4 \end{vmatrix}\]

Step 3: Expand 3x3 determinants

Calculate the determinants of each 3x3 matrix using the formula for a 3x3 determinant:- For the first cofactor (\(-2\)): \(6(0*0 - 5*4) - (-2)(7*0 - 5*(-12)) + 3(7*4 - 7*(-12))\) Simplifying, the determinant is \(-120 + 120 + 252 = 252\).- For the second cofactor (\(3\)): \(4(0*0 - 5*4) - (-2)(7*0 - 5*3) + 3(7*4 - 7*3)\) Simplifying, the determinant is \(-80 + 30 + 28 = -22\).- For the third cofactor (\(-1\)): \(4(0*5 - 5*(-12)) - 6(7*3 - 5*3) + 3(7*(-12) - (-12)*7)\) Simplifying, the determinant is \(240 - 36 + 0 = 204\).- For the fourth cofactor (\(7\)): \(4(7*4 - 0*(-12)) - 6(3*0 - 4*3) - 2(7*(-12) - 7*3)\) Simplifying, the determinant is \(112 + 24 + 42 = 178\).

Step 4: Combine cofactor results

Now, combine the cofactor results using the formula for the determinant of a 4x4 matrix:\[-2(252) - 3(-22) - 1(204) + 7(178)\]Compute the values:\(-504 + 66 - 204 + 1246 = 604\).

Step 5: Final Answer

The determinant of the given 4x4 matrix is 604.

Key Concepts

Cofactor Expansion

Cofactor expansion is a method used in linear algebra to compute the determinant of a square matrix. To understand the idea image it like peeling layers from an onion. Rather than computing the determinant of a big matrix all at once, we simplify it by breaking it down into smaller matrices. You can expand along any row or column, and for each element in that row or column, you calculate something called a cofactor.
Here’s how it works:

• Pick a row or column for expansion. In our example, we chose the first row.
• For each element in the chosen row or column, calculate its cofactor:
  • The cofactor is obtained by removing the row and column containing that element and finding the determinant of the resulting smaller matrix.
• Multiply each cofactor by the original element and add or subtract it from your total, alternating signs, starting with positive for the first element.

This approach leverages simpler computations of smaller matrices and produces the determinant of the original matrix. Cofactor expansion reveals the recursive nature of determinants.

Matrix Operations

Matrix operations are fundamental techniques in linear algebra used to simplify calculations, including determinant evaluation.
Key operations you should know include:

• Row and Column Manipulations:
  These include swapping rows or columns, multiplying a row or column by a scalar, or adding a multiple of one row to another row. These operations adjust the matrix without changing certain properties like solvability of equations.


• Determinant Properties:
  • Swapping two rows or columns changes the sign of the determinant.
  • Multiplying a row by a scalar multiplies the determinant by that scalar.
  • Adding a multiple of one row to another does not change the determinant.
• Use of Identity Matrix: The identity matrix helps in maintaining the properties during operations and is often used in simplifying larger matrix problems.

Using matrix operations strategically can simplify complex matrices, making determinant evaluation much more manageable.

4x4 Matrix Determinant

The determinant of a 4x4 matrix can seem overwhelming at first, but it's just a matter of applying the techniques systematically.
When solving determinants of this size, you start by identifying the method that will simplify the computation, such as cofactor expansion.
In our specific example of a 4x4 matrix, we:

• Chose the first row for expansion, given its relatively simpler numbers or zeroes that can ease calculations.
• Calculated the cofactors for each element in this row by forming three 3x3 matrices.
• Simplified each 3x3 matrix by finding its determinant, stepping through smaller determinant calculations.
• Ultimately combined these results to find the overall determinant of the matrix.

The overall determinant calculation functioned as a step-by-step teardown of a more significant problem into bite-sized, solvable pieces.
This process emphasizes the utility of methodically applying matrix operations and determinant properties to efficiently compute the solution.