Question

Find the determinant of the given matrix using cofactor expansion along any row or column you choose. \(\left[\begin{array}{ccc}1 & 2 & 3 \\ -5 & 0 & 3 \\ 4 & 0 & 6\end{array}\right]\)

Short answer

The determinant of the matrix is 84.

Step-by-step solution

Choose a Row or Column for Expansion

We will use cofactor expansion along the first row for calculating the determinant of the matrix. This choice is convenient as the first row contains a zero, which simplifies calculations.

Calculate the Cofactor of First Element

For the element \(1\) in the first row, calculate its cofactor. The cofactor is determined by the minor of the matrix obtained by removing the first row and first column: \[ \begin{vmatrix} 0 & 3 \ 0 & 6 \end{vmatrix} = (0)(6) - (3)(0) = 0 \]Multiply by the original element and sign: \(1 \times (-1)^{1+1} \times 0 = 1 \times 0 = 0\).

Calculate the Cofactor of Second Element

For the element \(2\), the minor is obtained by removing the first row and second column:\[ \begin{vmatrix} -5 & 3 \ 4 & 6 \end{vmatrix} = (-5)(6) - (3)(4) = -30 - 12 = -42 \]Multiply the original element by its cofactor with the sign adjustment:\(2 \times (-1)^{1+2} \times (-42) = -2 \times (-42) = 84\).

Calculate the Cofactor of Third Element

For the element \(3\), remove the first row and the third column to find the minor:\[ \begin{vmatrix} -5 & 0 \ 4 & 0 \end{vmatrix} = (-5)(0) - (0)(4) = 0 \]Calculate the contribution by multiplying:\(3 \times (-1)^{1+3} \times 0 = 3 \times 0 = 0\).

Sum the Cofactor Products

Add the results from each cofactor calculation to find the determinant:\(0 + 84 + 0 = 84\). Therefore, the determinant of the matrix is \(84\).

Key concepts

Cofactor Expansion

Cofactor expansion is a technique used to compute the determinant of a square matrix. It involves "expanding" the determinant along a chosen row or column. By computing the determinant this way, calculations become manageable, especially when zeros are present. The determinant is simplified by considering each element of the row (or column) and multiplying it by its cofactor.

• First, select one row or column to expand. You can choose any row or column, but picking one with zeros can simplify your arithmetic.
• Next, compute the cofactor for each element in the row or column. The cofactor is calculated as the product of the element, the minor, and the sign which alternates based on the position of the element in the matrix.
• Finally, add the products of all elements in the chosen row or column to get the determinant.

Cofactor expansion allows flexibility and efficiency when tackling large matrices, especially when zeros are carefully chosen to minimize calculations.

Matrix Algebra

Matrix algebra involves operations and rules that apply to matrices, which are arrays of numbers arranged in rows and columns. One fundamental concept in matrix algebra is the determinant, which can reveal information about the matrix, such as whether it has an inverse.

• To work with matrices, you need to be familiar with concepts such as matrix addition, subtraction, and multiplication. These operations follow specific mathematical rules similar to those of regular numbers.
• However, unlike numbers, matrix multiplication is not commutative; meaning, the order in which you multiply matrices matters!
• The determinant is particularly useful because it determines whether a matrix is invertible. A non-zero determinant indicates that the matrix can be inverted, making the determinant a critical aspect of matrix algebra.

Understanding these basics allows you to apply more advanced techniques and solve various problems involving matrices.

Minor of a Matrix

The minor of a matrix is a vital concept within the determinant calculation. It is formed by deleting one row and one column from the original matrix. The resulting smaller matrix is used to calculate the determinant in cofactor expansion.

• To find the minor of an element in a matrix, remove the row and column that contain that element. The matrix left is the minor of that element.
• The minor is then used to compute the cofactor. This involves taking the determinant of the minor and adjusting it with a sign based on the element's position.
• Minors reduce a complex problem to smaller, manageable pieces, which are easier to handle when calculating the determinant of a larger matrix.

This reduction to minors when calculating determinants underpins the method of cofactor expansion, making them essential in simplifying and solving matrix algebra problems.