Problem 32 - Algebra for College Students

Question

Evaluate each \(4 \times 4\) determinant. Use the properties of determinants to your advantage. \(\left|\begin{array}{rrrr}1 & 2 & 0 & 0 \\ 3 & -1 & 4 & 5 \\ -2 & 4 & 1 & 6 \\\ 2 & -1 & -2 & -3\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant is -152.

Step 1: Define the Determinant

The determinant of the matrix is given as: \[\left|\begin{array}{rrrr}1 & 2 & 0 & 0 \3 & -1 & 4 & 5 \-2 & 4 & 1 & 6 \2 & -1 & -2 & -3\end{array}\right|\] Our task is to evaluate this 4x4 determinant.

Step 2: Simplifying by Row Operations

We notice that the first row contains two zeros. We can simplify the calculation by using cofactor expansion along the first row, which will reduce the computation required since any terms with zero will vanish.

Step 3: Cofactor Expansion of First Row

Expand the determinant along the first row:\[1 \cdot \left|\begin{array}{ccc}-1 & 4 & 5 \4 & 1 & 6 \-1 & -2 & -3\end{array}\right|- 2 \cdot \left|\begin{array}{ccc}3 & 4 & 5 \-2 & 1 & 6 \2 & -2 & -3\end{array}\right|\]

Step 4: Evaluate 3x3 Determinant 1

For the first 3x3 determinant:\[\left|\begin{array}{ccc}-1 & 4 & 5 \4 & 1 & 6 \-1 & -2 & -3\end{array}\right|= -1\left(1(-3) - 6(-2)\right) - 4\left(4(-3) - 6(-1)\right) + 5\left(4(-2) - 1(-1)\right)\]Simplify each term and sum them up:For the first term: \[-1((-3) + 12) = -1(9) = -9\]For the second term: \[-4(-12 + 6) = -4(-6) = 24\]For the third term: \[5(-8 - 1) = 5(-9) = -45\]So, the determinant is:\[-9 + 24 - 45 = -30\].

Step 5: Evaluate 3x3 Determinant 2

For the second 3x3 determinant:\[\left|\begin{array}{ccc}3 & 4 & 5 \-2 & 1 & 6 \2 & -2 & -3\end{array}\right|= 3\left(1(-3) - (-2)(6)\right) - 4\left(-2(-3) - 2(6)\right) + 5\left(-2(-2) - 2(1)\right)\]Simplify each term and sum them up:For the first term: \[3(-3 + 12) = 3(9) = 27\]For the second term: \[-4(6 - 12) = -4(-6) = 24\]For the third term: \[5(4 - 2) = 5(2) = 10\]So, the determinant is:\[27 + 24 + 10 = 61\].

Step 6: Substitute Back into Original Expression

Substitute the values back into the expression expanded in Step 3:\[1(-30) - 2(61)\]Calculate the result:\[-30 - 122 = -152\].

Step 7: Final Result of the Determinant

The value of the 4x4 determinant is \[-152\].

Key Concepts

Cofactor Expansion

Cofactor expansion, also referred to as Laplace's expansion, is a method to compute the determinant of a matrix. Imagine peeling layers off a complex problem, solving simpler parts, and then putting everything back together. This technique involves expressing the determinant of a matrix as a sum of determinants of its smaller submatrices, called minors.

Here's how it works: choose any row or column of the matrix. For each element in that row or column, calculate the determinant of the submatrix obtained by eliminating the row and column of that element. This submatrix determinant is known as the minor. Combine each minor with a sign based on its position, resulting in a cofactor. Sum these cofactors to get the original matrix's determinant.

An essential tip: choose a row or column with the most zero entries to simplify calculations. In our exercise, we expanded along the first row of the matrix, which contained zeros, making the process quicker and more efficient.

Matrix Row Operations

Matrix row operations are powerful tools used to simplify matrices and help us find their determinant more efficiently. There are three main types of row operations:

• Swapping rows: Changing the order of rows doesn't affect the determinant unless you swap two rows, which changes the sign of the determinant.
• Multiplying a row by a scalar: If you multiply a row by a constant, the determinant is also multiplied by that constant.
• Adding or subtracting rows: Adding a multiple of one row to another row keeps the determinant unchanged.

Using these operations, we can transform a matrix to make calculating its determinant easier, such as forming zeros in ideal positions for cofactor expansion. In the given problem, the first row already had zeros, allowing us to jump straight into cofactor expansion without extra row manipulation.

4x4 Matrix Calculus

Evaluating the determinant of a 4x4 matrix may seem daunting, but it's manageable by breaking the process into smaller parts. Understanding the concepts of cofactor expansion and matrix row operations helps demystify it.

A 4x4 matrix determinant involves choosing a row or column for expansion. In our example, using the first row with zeros strategically simplified calculations. This approach broke the 4x4 determinant into smaller 3x3 determinants.

To evaluate a 3x3 determinant, further expand into sum of its 2x2 determinants using cofactor expansion again. Identify smaller 2x2 blocks and compute their determinants, then combine these into the 3x3 result. Finally, complete the calculation for the full 4x4 determinant using these results.

Practice is key. With each step, you'll become more familiar with patterns, ultimately improving your efficiency when evaluating larger matrices.