Problem 4 - Precalculus: Mathematics for Calculus

Question

Fill in the blanks with appropriate numbers to calculate the determinant. Where there is "I", choose the appropriate sign \((+\text { or }-)\) $$\text { (a) }\left|\begin{array}{rr} 2 & 1 \\ -3 & 4 \end{array}\right|=$$ $$\text { (b) }\left|\begin{array}{rrr} 1 & 0 & 2 \\ 3 & 2 & 1 \\ 0 & -3 & 4 \end{array}\right|=$$

Step-by-Step Solution

Answer

The determinants are (a) 11 and (b) -7.

Step 1: Identify Matrix Type

Determine whether each matrix is a 2x2 or 3x3 matrix. The matrices provided are \(2 \times 2\) and \(3 \times 3\) respectively.

Step 2: Calculate the Determinant of the 2x2 Matrix

For matrix (a): \( \begin{pmatrix} 2 & 1 \ -3 & 4 \end{pmatrix} \), use the determinant formula for a 2x2 matrix: if \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), then the determinant is calculated as: \( (a \cdot d) - (b \cdot c) \). Substitute \( a = 2\), \( b = 1\), \( c = -3\), and \( d = 4 \), giving: \( (2 \cdot 4) - (1 \cdot (-3)) = 8 + 3 = 11 \).

Step 3: Expand the 3x3 Matrix Using the First Row

For matrix (b): \( \begin{pmatrix} 1 & 0 & 2 \ 3 & 2 & 1 \ 0 & -3 & 4 \end{pmatrix} \), use the cofactor expansion along the first row. The general formula for expansion is \( a_{11} \cdot C_{11} + a_{12} \cdot C_{12} + a_{13} \cdot C_{13} \), where \( C \) are the cofactors. Substituting in the values, we have: \( 1 \cdot \left|\begin{array}{cc} 2 & 1 \ -3 & 4 \end{array}\right| - 0 \cdot \left|\begin{array}{cc} 3 & 1 \ 0 & 4 \end{array}\right| + 2 \cdot \left|\begin{array}{cc} 3 & 2 \ 0 & -3 \end{array}\right| \).

Step 4: Calculate Each 2x2 Determinant Inside the Expansion

Calculate \( \left|\begin{array}{cc} 2 & 1 \ -3 & 4 \end{array}\right| \) as done in Step 2, which is \( 11 \). Next, calculate \( \left|\begin{array}{cc} 3 & 2 \ 0 & -3 \end{array}\right| \). Use the formula: \( (3 \cdot (-3)) - (2 \cdot 0) = -9 \).

Step 5: Combine Results with Appropriate Signs

Combine the results from Step 4: \( 1 \cdot 11 - 0 \cdot 0 + 2 \cdot (-9) = 11 + 0 - 18 = -7 \). Therefore, the determinant of matrix (b) is -7.

Key Concepts

Understanding the 2x2 Matrix Determinant

Calculating the determinant of a 2x2 matrix is a fundamental skill in linear algebra. A 2x2 matrix looks like this: \[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \]. To find its determinant, use the simple formula: \[ (a \cdot d) - (b \cdot c) \]. This formula results from subtracting the product of the top-left and bottom-right elements from the product of the top-right and bottom-left elements.

In this exercise, consider the 2x2 matrix presented as \( \begin{pmatrix} 2 & 1 \ -3 & 4 \end{pmatrix} \). Following our determinant formula, we calculate it as: \[ (2 \cdot 4) - (1 \cdot (-3)) \]. Breaking it down:

• Multiply 2 and 4 to get 8.
• Multiply 1 and -3 to get -3.
• Subtract these results: 8 - (-3) = 8 + 3 = 11.

Hence, the determinant of this 2x2 matrix is 11.

Remember, determinants help in matrix inversion and solve systems of linear equations, making them crucial in practical applications.

Calculating the 3x3 Matrix Determinant

Dealing with a 3x3 matrix can sound intimidating, but with the right steps, it becomes manageable. A 3x3 matrix has the form: \[ \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \]. To calculate its determinant, you generally use the cofactor expansion method. This involves expanding along any row or column (usually the first row for simplicity).

For our example matrix \[ \begin{pmatrix} 1 & 0 & 2 \ 3 & 2 & 1 \ 0 & -3 & 4 \end{pmatrix} \], we expand along the first row. The formula becomes: \[ a_{11} \cdot C_{11} + a_{12} \cdot C_{12} + a_{13} \cdot C_{13} \], where each \( C \) is the minor determinant.

Work through each:

• The minor \( C_{11} \) is the determinant of \[ \begin{pmatrix} 2 & 1 \ -3 & 4 \end{pmatrix} \] which is 11 (already calculated).
• The minor \( C_{12} \) would not affect the solution as \( a_{12} \) is 0, so it's omitted.
• The minor \( C_{13} \) derived from \[ \begin{pmatrix} 3 & 2 \ 0 & -3 \end{pmatrix} \] calculates as \[ (3 \cdot (-3)) - (2 \cdot 0) = -9 \].

Finally, plug these into the formula: \[ 1 \cdot 11 + 0 \cdot 0 + 2 \cdot (-9) = 11 - 18 = -7 \]. Thus, the determinant of this 3x3 matrix is -7.

Mastering Cofactor Expansion

Cofactor expansion is a powerful tool to assess the determinant of larger matrices, such as a 3x3 matrix. It breaks down the problem into manageable pieces, using smaller matrices and simpler arithmetic.

Every element in a matrix can be associated with a minor and a sign, determined by its position. The general sign pattern forms a checkerboard of plus and minus, starting with a plus in the top-left corner.

Here's how it generally works:

• Select a row (or column) to expand along. First row is often chosen for simplicity.
• For each element, except columns with zero values (helps to simplify calculation), find the determinant of the 2x2 minor matrix that remains after removing the row and column of the element.
• Multiply each minor determinant by the corresponding matrix element.
• Apply the sign pattern \( (+,-,+) \) for the row elements \( a_{11}, a_{12}, a_{13} \).
• Accumulate the results according to the signs.

This method not only helps in calculating the determinant, but also builds a deeper understanding of matrix structures. Always remember that practice makes perfect, and breaking problems into smaller parts is key to mastering them.