Problem 9 - Intermediate Algebra

Question

Evaluate each of the following determinants. $$ \left|\begin{array}{rr} -3 & 2 \\ 5 & -6 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 8.

Step 1: Understanding the Determinant

A determinant is a special number that can be calculated from a square matrix. In this case, we have a 2x2 matrix. The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is given by \( ad - bc \).

Step 2: Identify Elements of the Matrix

Identify the elements of the matrix \( \begin{pmatrix} -3 & 2 \ 5 & -6 \end{pmatrix} \). Here, \( a = -3 \), \( b = 2 \), \( c = 5 \), and \( d = -6 \).

Step 3: Apply the Determinant Formula

Substitute these values into the determinant formula \( ad - bc \). This gives: \( (-3)(-6) - (2)(5) \).

Step 4: Perform the Multiplications

Calculate each multiplication: \((-3) \times (-6) = 18\) and \(2 \times 5 = 10\).

Step 5: Subtract the Products

Finally, subtract the second product from the first: \(18 - 10 = 8\).

Key Concepts

2x2 matrix

A 2x2 matrix is a simple matrix form in linear algebra with two rows and two columns. It is often written as \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \). The elements \(a, b, c, \) and \(d\) are numbers that can be either real or complex. This type of matrix is fundamental in many areas of mathematics because it is the smallest size matrix from which one can compute a determinant.

• The first row consists of elements \(a\) and \(b\).
• The second row has elements \(c\) and \(d\).

Understanding how to find determinants of 2x2 matrices is important because it forms the basis for working with larger matrices and matrix-related operations, such as finding inverse matrices.

matrix multiplication

Matrix multiplication is a way to combine two matrices to produce a new matrix. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second. While not directly related to calculating determinants, understanding multiplication is essential for grasping how matrices interact in equations.

For example, if you have two 2x2 matrices, \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) and \( B = \begin{pmatrix} e & f \ g & h \end{pmatrix} \), their product is another 2x2 matrix given by:\[AB = \begin{pmatrix} ae + bg & af + bh \ ce + dg & cf + dh \end{pmatrix}\]
This process involves multiplying the rows of the first matrix by the columns of the second.

linear algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear transformations, and systems of linear equations. A key aspect of linear algebra is the study of matrices and determinants, which helps in solving various problems related to these topics.

In linear algebra, a determinant gives us information about a matrix's properties, such as:

• Whether the matrix is invertible
• The volume scaling factor for linear transformations represented by the matrix
• Properties related to matrix eigenvalues

Determinants are also utilized when solving systems of linear equations using methods like Cramer's Rule. Overall, linear algebra provides a powerful framework for analyzing and solving complex mathematical problems.

matrices

Matrices are rectangular arrays of numbers arranged in rows and columns. They are a fundamental concept in linear algebra and are used to represent linear transformations, solve systems of linear equations, and perform various computations. The size or dimension of a matrix is defined by the number of its rows and columns.

Some common uses of matrices include:

• Representing data in fields like economics, statistics, and physics
• Describing linear transformations in graphics and engineering
• Formulating and solving systems of linear equations

Matrices are versatile tools that appear in many areas of mathematics and applied science, making them essential for students to understand. They are often processed using row operations, matrix multiplication, and determinants to extract useful information.