Question

Find the determinant of the \(2 \times 2\) matrix. \(\left[\begin{array}{cc}6 & -1 \\ -7 & 8\end{array}\right]\)

Short answer

The determinant of the matrix is 41.

Step-by-step solution

Identify Matrix Elements

For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), identify elements \(a\), \(b\), \(c\), and \(d\). In this matrix, \(a = 6\), \(b = -1\), \(c = -7\), and \(d = 8\).

Apply the Determinant Formula for 2x2 Matrix

For a 2x2 matrix, the determinant is calculated as \(ad - bc\). Substitute the identified values: \(a = 6\), \(b = -1\), \(c = -7\), \(d = 8\) into the formula.

Calculate the Products

Calculate the product \(ad = 6 \times 8 = 48\). Then calculate the product \(bc = (-1) \times (-7) = 7\).

Compute the Determinant

Subtract the products: \(ad - bc = 48 - 7 = 41\).

State the Result

The determinant of the matrix \(\begin{bmatrix} 6 & -1 \ -7 & 8 \end{bmatrix}\) is 41.

Key concepts

2x2 Matrix

A \(2 \times 2\) matrix is a simple, square matrix that is commonly used in mathematics to perform various operations. This matrix consists of two rows and two columns, which generates a total of four elements. These elements can be numbers, variables, or expressions. For example, in a matrix represented as:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]each element has a specific position:

• \(a\) is located at the first row, first column
• \(b\) is located at the first row, second column
• \(c\) is located at the second row, first column
• \(d\) is located at the second row, second column

Understanding the structure of a \(2 \times 2\) matrix is essential for performing operations like addition, subtraction, and, in this case, finding the determinant. The setup is straightforward, which makes it a great starting point for learning matrix algebra.

Determinant Calculation

Calculating the determinant of a matrix is a key concept in linear algebra, particularly for square matrices. For a \(2 \times 2\) matrix, the determinant is a simple yet powerful calculation that provides meaningful insights into the matrix's properties.To calculate the determinant of the matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), we use the formula:\[\text{Det}(M) = ad - bc\]Here is how it works for our example matrix \(\begin{bmatrix} 6 & -1 \ -7 & 8 \end{bmatrix}\):

• Identify the elements: \(a = 6\), \(b = -1\), \(c = -7\), \(d = 8\)
• Compute the product \(ad = 6 \times 8 = 48\)
• Compute the product \(bc = (-1) \times (-7) = 7\)
• Subtract the two products: \(48 - 7 = 41\)

Thus, the determinant of this matrix is \(41\). This process demonstrates the simplicity and efficiency of determinant calculation for a \(2 \times 2\) matrix.

Matrix Algebra

Matrix algebra involves operations and manipulations of matrices, which are essential tools in various fields such as mathematics, physics, and engineering. While matrices can be of any dimension, understanding simple matrix operations and properties using \(2 \times 2\) matrices forms the foundation of broader matrix theory.Key operations in matrix algebra include:

• Addition and Subtraction: Performed by adding or subtracting corresponding elements of matrices of the same dimension.
• Multiplication: Two matrices can be multiplied if the number of columns in the first matrix matches the number of rows in the second matrix. This operation is more complex than it sounds and has specific rules.
• Inverse: The inverse of a matrix, if it exists, can "undo" the effects of matrix multiplication. The determinant plays a critical role in determining whether a matrix has an inverse. If the determinant is zero, the matrix does not have an inverse.

Understanding these operations enables solving systems of equations, transforming geometric figures, and more. Matrix algebra is not just limited to solving theoretical problems but is also heavily applied in real-world scenarios, making its study both practical and essential.