Question

Find the determinant of the given matrix using cofactor expansion along any row or column you choose. \(\left[\begin{array}{ccc}0 & -3 & 1 \\ 0 & 0 & 5 \\ -4 & 1 & 0\end{array}\right]\)

Short answer

The determinant of the matrix is 60.

Step-by-step solution

Select a Row or Column for Expansion

Choose a row or column to perform the cofactor expansion. It's often easier to select a row or column with zeros to reduce the number of calculations. Here, choose the first column, which includes two zeros.

Identify Cofactors

For the chosen column (first column), identify the elements and their positions: - Element: 0 at position (1,1) - Element: 0 at position (2,1) - Element: -4 at position (3,1). The formula for a cofactor is \[ C_{ij} = (-1)^{i+j} M_{ij} \] where \( M_{ij} \) is the minor of element at \( (i,j) \).

Calculate Minors

Calculate the minors (\( M_{ij} \)) for each element in the first column that is non-zero: - For element -4 at position (3,1), the minor is the determinant of the matrix obtained by removing the 3rd row and 1st column: \[ \begin{vmatrix} -3 & 1 \ 0 & 5 \end{vmatrix} \].

Compute Sub-Determinants

Calculate the determinant of the 2x2 matrix \( \begin{vmatrix} -3 & 1 \ 0 & 5 \end{vmatrix} \) by cross-multiplying:\[ (-3)(5) - (1)(0) = -15 \].

Apply Cofactor Signs and Compute Determinant

Using the cofactor expansion method, sum the products of the elements and their respective cofactors. Here, only the element -4 contributes, as others are zero: For -4 at (3,1):\((-1)^{3+1} imes (-4) imes (-15) = 1 imes (-4) \times (-15) = 60\). Hence, the determinant is 60.

Key concepts

Cofactor Expansion

The cofactor expansion, also known as Laplace expansion, is a method to calculate the determinant of a matrix. It involves expanding a matrix along a selected row or column. Choose a row or column with many zeros; this simplifies calculations, as zero elements result in zero contributions to the determinant when expanded.

• The formula for cofactor expansion is: \[ ext{det}(A) = ext{sum}(a_{ij} C_{ij})\]where \( a_{ij} \) is an element from the chosen row or column, and \( C_{ij} \) is the cofactor of that element.
• Cofactors are calculated using \( C_{ij} = (-1)^{i+j} M_{ij} \), where \( M_{ij} \) is the minor for element \( a_{ij} \).

In practical terms, choosing a row or column with zeros minimizes the computational workload. This step is crucial in finding the determinant efficiently, especially in larger matrices where calculations can become complex.

Minor of a Matrix

The minor of a matrix element \( a_{ij} \) is the determinant of the submatrix formed by deleting the \(i\)-th row and \(j\)-th column in which the element appears. This smaller matrix is key to calculating cofactors, which are essential in methods like cofactor expansion.

• For element \( a_{ij} \), the minor \( M_{ij} \) is a reduced matrix, simplifying the larger matrix's determinant calculation by dealing with smaller matrices.
• For instance, in our example, the element \[-4\] at position \( (3,1) \) involved calculating the determinant of the submatrix \[\begin{vmatrix} -3 & 1 \ 0 & 5 \end{vmatrix}\]which is itself a simple 2x2 determinant.

Understanding minors simplifies the task of finding larger determinants since they allow breaking the problem down into easier steps, each tackling smaller components of the original matrix.

2x2 Determinant Calculation

Calculating the determinant of a 2x2 matrix is straightforward and forms the core element of dealing with larger matrices when using cofactor expansion. It's governed by a simple formula that involves multiply and subtract operations.

• The formula for a 2x2 matrix \( \begin{vmatrix} a & b \ c & d \end{vmatrix} \) is:\[ad - bc\]
• This calculation is easy to perform and is often used when determining minors in larger matrices.
• In our example, the determinant for matrix\[\begin{vmatrix} -3 & 1 \ 0 & 5 \end{vmatrix}\] was determined by calculating \[-3\times5 - 1\times0 = -15\].

This simple operation is a building block for tackling more complex matrices, allowing for efficient calculation by breaking down the process into manageable parts. Mastering the 2x2 determinant is key in understanding larger matrix computations.