Question

Find the determinant of the given matrix using cofactor expansion along the first row. \(\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 1 & 1\end{array}\right]\)

Short answer

The determinant is 1.

Step-by-step solution

Identify the Matrix Elements

The matrix we are given is \[\begin{bmatrix}1 & 0 & 0 \0 & 1 & 0 \-1 & 1 & 1\end{bmatrix}\]We will perform cofactor expansion along the first row. The elements of the first row are 1, 0, and 0.

Select the Element for Expansion

For cofactor expansion along the first row, we start with the first element, which is 1.

Calculate the Minor and Cofactor of the First Element

For the element 1 in the first row and first column, remove the first row and first column, giving the submatrix:\[\begin{bmatrix}1 & 0 \1 & 1\end{bmatrix}\]The determinant of this submatrix (the minor) is calculated as \(1 \times 1 - 0 \times 1 = 1\).The cofactor is then \((-1)^{1+1} \times 1 = 1\).

Calculate the Minor and Cofactor of the Second Element

For the element 0 in the first row and second column, remove the first row and second column:\[\begin{bmatrix}0 & 0 \-1 & 1\end{bmatrix}\]The determinant of this submatrix (the minor) is \(0 \times 1 - 0 \times -1 = 0\).The cofactor is \((-1)^{1+2} \times 0 = 0\).

Calculate the Minor and Cofactor of the Third Element

For the element 0 in the first row and third column, remove the first row and third column:\[\begin{bmatrix}0 & 1 \-1 & 1\end{bmatrix}\]The determinant of this submatrix (the minor) is \(0 \times 1 - 1 \times -1 = 1\).The cofactor is \((-1)^{1+3} \times 1 = 1\).

Apply Cofactor Expansion Formula

The determinant by cofactor expansion along the first row is calculated as\(1 \times 1 + 0 \times 0 + 0 \times 1 = 1\).

Final Determinant Result

After calculating the cofactor sums, the determinant of the matrix is 1.

Key concepts

Cofactor Expansion

Cofactor expansion is a method used for calculating the determinant of a matrix. This technique involves expanding the determinant along a chosen row or column. For each element in the row or column, a specific value called the _cofactor_ is calculated and then summed to determine the entire matrix's determinant. Here's how cofactor expansion works:

• Choose a row or column to expand along. Typically, the row or column with the most zeros is a good choice because it simplifies calculations.
• For each element in the chosen row or column, calculate the cofactor, which involves finding the determinant of a smaller matrix (a "minor") and adjusting it with a sign based on the position of the element.

This method is a systematic way to break down complex matrices into smaller, manageable pieces, ultimately simplifying determinant calculations.

Matrix Algebra

Matrix algebra is a foundational concept in linear algebra which involves various operations, like addition, subtraction, and multiplication, performed on matrices. Determinants are a core subject within this field. When discussing matrix algebra, determinants help in solving system of equations, understanding matrix properties, and performing transformations. A determinant is a scalar value that can be computed from the elements of a square matrix and describes certain properties of the matrix, including:

• If the matrix is invertible (a non-zero determinant means it is invertible).
• The volume change factor when a linear transformation represented by the matrix is applied.

Determinants play a crucial role in various applications such as calculating volumes, solving linear equations, and even in computer graphics for transformations. Knowing how to evaluate them is vital for understanding and using matrices effectively.

Minor and Cofactor

In the process of cofactor expansion, you'll frequently encounter terms like _minor_ and _cofactor_. Understanding these will make matrix determinant calculations much more intuitive.**Minor:**

• A minor is the determinant of a smaller matrix formed by removing one row and one column from a larger matrix.
• To find the minor of an element, you discard its row and column, forming a submatrix.

**Cofactor:**

• The cofactor of an element is its minor, multiplied by \((-1)^{i+j}\), where \(i\) is the row number and \(j\) is the column number of the element.
• This multiplication by \((-1)^{i+j}\) helps to account for the orientation of the matrix axes.

Every element in a matrix can have a minor and cofactor associated with it, forming the basis for calculating the determinant through cofactor expansion.

Submatrix Extraction

Submatrix extraction is a critical step in finding the minor of an element during cofactor expansion. It involves isolating a smaller matrix from the original by removing one row and one column. Steps in submatrix extraction:

• Identify the element whose minor you need to calculate.
• Remove the row and column of this element from the larger matrix.
• What's left is the submatrix.

Submatrix extraction simplifies the problem by focusing on a smaller matrix, making complex determinant calculations more manageable. This smaller part or submatrix is used to calculate the minor, which is integral to finding the complete determinant of the original matrix.