Question

Use Laplace expansion to find the determinant of \(A=\left[\begin{array}{rrr}15 & 7 & 9 \\ 2 & 5 & 6 \\ 9 & 0 & 12\end{array}\right]\)

Short answer

The determinant of matrix \(A\) is 705.

Step-by-step solution

Choose a Row or Column

To use the Laplace expansion, you can choose any row or column. We will choose the first row for this example, which is \([15, 7, 9]\). This is because it doesn't contain zero elements, and it's often easier to use the first row.

Calculate the Cofactors

For each element in the first row, calculate the cofactor. The cofactor is determined by the formula \(C_{ij} = (-1)^{i+j}M_{ij}\) where \(M_{ij}\) is the determinant of the submatrix that remains after removing the \(i\)-th row and \(j\)-th column.- For \(15\), the submatrix is \( \begin{bmatrix} 5 & 6 \ 0 & 12 \end{bmatrix} \). The determinant is \(5 \times 12 - 6 \times 0 = 60\).- For \(7\), the submatrix is \( \begin{bmatrix} 2 & 6 \ 9 & 12 \end{bmatrix} \). The determinant is \(2 \times 12 - 6 \times 9 = 24 - 54 = -30\).- For \(9\), the submatrix is \( \begin{bmatrix} 2 & 5 \ 9 & 0 \end{bmatrix} \). The determinant is \(2 \times 0 - 5 \times 9 = 0 - 45 = -45\).

Apply Laplace Expansion Formula

Using the cofactors calculated in Step 2, apply the Laplace expansion formula along the first row: \[ \det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} \].Substitute the values: \[ \det(A) = 15 \times 60 - 7 \times (-30) + 9 \times (-45) \].

Simplify the Expression

Perform the arithmetic to simplify the expression:- Calculate \(15 \times 60 = 900\).- Calculate \(-7 \times (-30) = 210\).- Calculate \(9 \times (-45) = -405\).Add these results together to find the determinant:\[ 900 + 210 - 405 = 705 \].

Key concepts

Laplace Expansion

Laplace expansion, also known as cofactor expansion, is a technique used to compute the determinant of a square matrix. It involves expanding the determinant along a row or column. This is especially useful when the matrix is larger than 2x2. In this method, you select a row or column, then express the determinant as a sum of elements from that row or column, multiplied by their corresponding cofactors.

• Choose any row or column to perform the expansion. Ideally, select the one with the most zeros for simplicity.
• Calculate the cofactor of each element in the chosen row or column.
• Use the formula for Laplace expansion to find the determinant.

By following these steps, you can systematically reduce the complexity of the problem, transforming a large matrix determinant in more manageable parts.

Cofactor

The concept of a cofactor is integral when using Laplace expansion. It significantly influences the calculation of the determinant. A cofactor is a signed number created from a minor, which is the determinant of a submatrix formed by deleting a specific row and column from the original matrix.

• Notation: Denoted as \( C_{ij} \), where \( i \) and \( j \) are the row and column indices.
• Calculation: Use the formula \( C_{ij} = (-1)^{i+j}M_{ij} \) where \( M_{ij} \) is the determinant of the minor matrix.
• Significance: The alternating +/- sign results directly from the \((-1)^{i+j}\) factor, which helps account for orientation in multivariable calculus.

Understanding cofactors involves understanding both the calculation and their role in determining the overall determinant of the matrix.

Matrix Algebra

Matrix algebra is a branch of mathematics involving matrices and their operations. It is fundamental because it provides the tools needed to compute determinants and execute operations like multiplication, inversion, and transposition with matrices. Determinants offer critical insight into matrix properties such as invertibility and solution sets in systems of linear equations.

• Foundation: Matrices are rectangular arrays of numbers, which can represent linear mappings.
• Operations: Include addition, subtraction, and multiplication, which can follow complex specific rules.
• Importance: Within Laplace expansion, understanding basic operations is crucial to simplifying each expression and performing accurate calculations.

Mastery of matrix algebra forms a crucial skill for further mathematical concepts and applications, crucially supporting determinant calculations.

Submatrix Determinant

Calculating the determinant of a submatrix is a vital step in determining the cofactor during Laplace expansion. A submatrix is whatever remains after removing a specific row and column from the larger matrix. Finding its determinant is straightforward when the submatrix is 2x2: simply apply the formula for 2x2 determinants.

• Creation: Derived by deleting a row and column of the matrix.
• Calculation Method: For a 2x2 submatrix, use \( ad - bc \) where the submatrix is \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \).
• Significance: Submatrix determinants directly influence the size of the cofactor and ultimately the major determinant of the matrix.

Understanding how to handle submatrices and their determinants is crucial for efficiently carrying out larger matrix operations.