Problem 4 - College Algebra and Calculus: An Applied Approach

Question

Find the determinant of the matrix. $$ \left[\begin{array}{ll} -3 & 4 \\ -2 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the matrix is 5.

Step 1: Identify the elements of the matrix

Look at the matrix and identify the elements. It is a 2x2 matrix, hence it has four elements. These are a = -3, b = 4, c = -2 and d = 1.

Step 2: Apply the formula for the determinant

Now apply the formula for the determinant of a 2x2 matrix, which is \(ad - bc\). Therefore, substitute the identified elements into the formula. This gives \((-3)*1 - 4*(-2)\).

Step 3: Calculate the determinant

The last step is to carry out the arithmetic operations to calculate the determinant. So, \((-3)*1 - 4*(-2) = -3 + 8 = 5\).

Key Concepts

2x2 matrix

A 2x2 matrix is a simple and essential building block in linear algebra. It is a rectangular array consisting of two rows and two columns, which makes a total of four elements. In mathematical notation, a 2x2 matrix is often written as: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \] Here, each element, such as \(a, b, c,\) and \(d\), represents a specific number. These numbers can be any real or complex values. Matrices like these are used widely in mathematics to solve linear equations, perform transformations, and more. They provide a concise way to represent data and operations in a structured format.

Matrix arithmetic

Matrix arithmetic involves performing operations on matrices in order to obtain new matrices or results. One of the fundamental operations is addition, where matrices of the same dimension are added by summing corresponding elements. Subtraction works in a similar way. When it comes to multiplication, matrices are not multiplied element-wise but by considering the row by column rule, which involves calculating the dot product between rows and columns. In the context of a 2x2 matrix, although we often deal with single matrices, understanding how they interact through these basic arithmetic operations is crucial. Matrix arithmetic helps in understanding transformations in various fields such as physics, engineering, and computer graphics.

Determinant calculation

The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix, the determinant is computed using the formula \( ad - bc \), where \( a, b, c, \) and \( d \) are the elements of the matrix. This formula is quite straightforward:

• Multiply the elements diagonally from the top left to the bottom right, that is, \( a \) and \( d \).
• Multiply the elements diagonally from the top right to the bottom left, which are \( b \) and \( c \).
• Subtract the second product from the first one.

The determinant provides insights into the properties of the matrix, such as whether it is invertible or not. For example, a matrix with a determinant of zero indicates that it is singular and not invertible. Calculating determinants is a fundamental skill in matrix theory and applications.