Question

Let \(A\) be a \(2 \times 2\) matrix; $$A=\left[\begin{array}{ll}a & b \\\c & d\end{array}\right]$$ Show why \(\operatorname{det}(A)=a d-b c\) by computing the cofactor expansion of \(A\) along the first row.

Short answer

The determinant is computed as \( \det(A) = ad - bc \) by cofactor expansion.

Step-by-step solution

Understanding the Determinant of a 2x2 Matrix

For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant \( \det(A) \) can be expressed as \( a \times d - b \times c \). We'll prove this using cofactor expansion along the first row.

Cofactor Expansion Definition

Cofactor expansion along a row or column expresses the determinant as a sum of products, where each product consists of an entry in the matrix and its corresponding cofactor. The cofactor \( C_{ij} \) is given by \( (-1)^{i+j} \) times the determinant of the minor matrix obtained by deleting the \(i\)-th row and \(j\)-th column.

Cofactor Expansion Along the First Row

For matrix \( A \), using cofactor expansion along the first row, the determinant is: \[ \det(A) = a \cdot C_{11} + b \cdot C_{12} \] where \( C_{11} = (-1)^{1+1} \cdot det(d) \) and \( C_{12} = (-1)^{1+2} \cdot det(c) \).

Calculate Cofactor \( C_{11} \)

The minor of element \( a \) is just the element \( d \) because we remove the first row and first column. Thus, the cofactor \( C_{11} = 1 \times d = d \).

Calculate Cofactor \( C_{12} \)

The minor of element \( b \) is just the element \( c \). Thus, the cofactor \( C_{12} = (-1) \times c = -c \).

Substitute Cofactors into the Expansion

Substitute the calculated cofactors back into the cofactor expansion:\[ \det(A) = a \cdot d + b \cdot (-c) \] which simplifies to \[ \det(A) = a \cdot d - b \cdot c \].

Key concepts

Cofactor Expansion

Cofactor expansion is a powerful technique that helps in finding the determinant of a matrix by breaking it down into more manageable parts. When performing cofactor expansion, you choose a row or column of the matrix. Then, for each element in the chosen row or column, you calculate its cofactor and multiply it with the element itself. The sum of these products gives you the determinant of the matrix.
The cofactor for an element is determined using the formula:

• The cofactor, denoted as \( C_{ij} \), is calculated as \((-1)^{i+j}\) times the determinant of the matrix’s minor, a smaller matrix formed by removing the \(i\)-th row and \(j\)-th column.

Understanding cofactor expansion is crucial because it not only simplifies the process of computing determinants, especially for larger matrices, but also deepens comprehension of how determinants work in matrix algebra.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices, their operations, and properties. Matrices, which are rectangular arrays of numbers or functions, represent and facilitate the handling of linear transformations.
In matrix algebra, several operations are critical:

• Addition: You add matrices element-wise, requiring both to have the same dimensions.
• Multiplication: Matrix multiplication involves each element of the resulting matrix being the dot product of corresponding row and column vectors.
• Determinant: Determinants of square matrices are scalar values that provide insights into the properties of the matrix, like invertibility.

Matrix algebra is foundational in solving systems of linear equations, transforming geometrical data, and various applications across statistics, physics, and computer science.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix, consisting of two rows and two columns. An example of a 2x2 matrix can be given by:\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]
Calculating the determinant of a 2x2 matrix is a straightforward process and serves as a fundamental exercise in understanding larger matrices' properties. The determinant of a 2x2 matrix is determined by the formula:\[ ext{det}(A) = ad - bc\]
This simple calculation helps to check whether the matrix is invertible. If the determinant is non-zero, the matrix has an inverse. Many essential matrix properties and operations can be explored using these simple 2x2 matrices before handling more complex and larger ones.

Minor

The minor of a matrix element is generated by removing the row and column that the element belongs to from the matrix. It's a smaller matrix, obtained as part of the process of calculating cofactors.
For a 2x2 matrix:

• Consider the matrix : \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \)
• Minor of \(a\): Remove the first row and first column. The minor of \(a\) is \([d]\).
• Minor of \(b\): Remove the first row and second column. The minor of \(b\) is \([c]\).

The minor plays a vital role in cofactor calculations, which in turn are essential for finding the determinant through cofactor expansion. Understanding minors is key to grasping the concept of how each element contributes to the determinant and why it affects the matrix's properties so significantly.