Question

Find the determinant of the given matrix using cofactor expansion along any row or column you choose. \(\left[\begin{array}{ccc}0 & 4 & -4 \\ 3 & 1 & -3 \\ -3 & -4 & 0\end{array}\right]\)

Short answer

The determinant is 72.

Step-by-step solution

Choose a Row or Column

To find the determinant using cofactor expansion, we can pick any row or column. Let's choose the first row for this example, which is \([0, 4, -4]\). This choice is often made because it contains a zero, which simplifies calculations.

Set Up the Cofactor Expansion

The determinant of the matrix will be calculated using the formula:\[ \text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} \]where \(a_{ij}\) are the elements of the row and \(C_{ij}\) are the cofactors.

Calculate Cofactors

For each non-zero element of the chosen row, calculate the minor and then the cofactor:- For \(a_{12} = 4\): - Remove the first row and second column to get the minor matrix \(\left[\begin{array}{cc}3 & -3 \ -3 & 0\end{array}\right]\) - Calculate the minor's determinant: \(3(0) - (-3)(-3) = -9\) - The cofactor \(C_{12} = (-1)^{1+2} \cdot (-9) = 9\)- For \(a_{13} = -4\): - Remove the first row and third column to get the minor matrix \(\left[\begin{array}{cc}3 & 1 \ -3 & -4\end{array}\right]\) - Calculate the minor's determinant: \(3(-4) - (-3)(1) = -9\) - The cofactor \(C_{13} = (-1)^{1+3} \cdot (-9) = -9\)

Compute the Determinant

Using the cofactor expansion formula from Step 2:\[ \text{det}(A) = 0 \cdot C_{11} + 4 \cdot 9 + (-4) \cdot (-9) \]\[ \text{det}(A) = 0 + 36 + 36 = 72 \]

Conclusion: Determinant Value

The determinant of the given matrix is 72.

Key concepts

Understanding Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a method used to calculate the determinant of a matrix.
The determinant is a special number that can indicate many properties of the matrix, such as whether it is invertible or not.
To perform a cofactor expansion, choose any row or column of the matrix. This is completely based on your convenience, though it's often helpful to choose a row or column with zeros to simplify calculations.
Once selected, you'll compute the determinant using the formula:

• Multiply each element of the chosen row or column by its corresponding cofactor.
• Add these products together.

The cofactor is determined by removing the elements' row and column from the matrix and finding the determinant of the resulting submatrix, flipping the sign as needed.
Thus, cofactor expansion turns a possibly complex process into manageable steps, allowing for easier computation of a determinant.

Exploring Matrix Minors

Matrix minors are a key component in cofactor expansion.
In mathematics, a minor of a matrix is the determinant of a smaller square matrix obtained by removing one row and one column from the original matrix.
In the context of a 3x3 matrix, such as our given example, a minor is derived from 2x2 matrices. Here's how you find a minor for a specific element:

• Remove the row and column that intersect at that specific element.
  For instance, to find the minor of element at position \(a_{12}\), eliminate the first row and second column.
• The remaining matrix will be a 2x2 matrix.

By calculating the determinant of this smaller matrix, you get the minor for the specific element.
The minor is crucial for finding the cofactor, where you'd apply a sign change based on the position's parity in the matrix using \((-1)^{i+j}\).

Working with a 3x3 Matrix

A 3x3 matrix, such as the one in this problem, is simply a matrix with three rows and three columns.
It is a common size for matrices, making it a staple in linear algebra exercises.Working with a 3x3 matrix involves understanding its structure:

• Each element in the matrix has a specific position denoted by two indices: the row number and the column number. This leads to terms like \(a_{ij}\) to identify elements.
• For things like cofactor expansion, it's helpful to visualize the matrix, especially when removing rows and columns to find minors.
• Matrix operations, such as finding determinants, can often be simplified by strategic selection of rows or columns, as some might contain zeros that ease the arithmetic.

Overall, a strong grasp of 3x3 matrices and how to manipulate them is essential for solving a wide variety of problems in mathematics and engineering.