Problem 9 - Precalculus: Functions and Graphs

Question

Find the determinant of the matrix. $$\left[\begin{array}{ll} -5 & 4 \\ -3 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is 2.

Step 1: Identify the matrix

We are given a 2x2 matrix:\[\left[ \begin{array}{cc}-5 & 4 \-3 & 2\end{array} \right] \]

Step 2: Recall the formula for a 2x2 determinant

The determinant of a 2x2 matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) is calculated using the formula:\[ \text{det}(A) = ad - bc \]

Step 3: Assign matrix elements to variables

In our matrix:- \( a = -5 \)- \( b = 4 \)- \( c = -3 \)- \( d = 2 \)

Step 4: Apply the determinant formula

Substitute the values into the determinant formula:\[ det(A) = (-5)(2) - (4)(-3) \]

Step 5: Calculate each term separately

First, calculate the product \((-5)(2)\):\[ (-5)(2) = -10 \]Then calculate \((4)(-3)\):\[ (4)(-3) = -12 \]

Step 6: Perform the determinant operation

Now, subtract the second term from the first:\[ -10 - (-12) = -10 + 12 = 2 \]

Step 7: State the determinant

The determinant of the matrix is:\[ det(A) = 2 \]

Key Concepts

2x2 matrix

A 2x2 matrix is a simple type of matrix used frequently in mathematics, especially in linear algebra. It consists of two rows and two columns, forming a square. Matrices are often denoted by brackets and have elements represented typically by letters, such as a matrix \( A \) with elements \( a, b, c, \) and \( d \):

• \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \)
• This matrix specifically resembles a coordinate system or grid, where each element can hold any real number value.

2x2 matrices are used for various computations such as transformations or solving systems of equations. They are also the foundation to understand larger matrices, allowing you to explore complex concepts in manageable segments. This concept is crucial for beginners in linear algebra, as starting with a 2x2 matrix provides the simplest non-trivial example to practice basic matrix operations like addition, multiplication, and finding determinants.

matrix determinant formula

The determinant is a scalar value that can be computed from the elements of a square matrix. It is crucial in matrix algebra for solving linear equations, finding inverses of matrices, and understanding matrix properties.

For a 2x2 matrix, the determinant is straightforward to calculate. The formula used is:

• \( ext{det}(A) = ad - bc \)

where \( A \) is the matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), and \( a, b, c, \) and \( d \) are the elements of the matrix.

To find the determinant, you multiply diagonally and subtract:

• First, multiply the elements of the leading diagonal: \( a \times d \)
• Then, multiply the elements of the opposite diagonal: \( b \times c \)
• Subtract the second product from the first, i.e., \( (ad) - (bc) \)

This calculation gives important insight into a matrix's properties, like whether it has an inverse (a non-zero determinant) or whether vectors in its system are linearly independent.

linear algebra concepts

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear mappings between these spaces. This field is essential for various applications across physics, computer science, and engineering.

At its core, linear algebra involves:

• Understanding and performing operations with matrices and vectors.
• Solving systems of linear equations.
• Investigating transformations, which can include rotation, scaling, or translation in space.

Matrix operations like addition, multiplication, and determinant computation form the basis of linear algebra's practical applications.

Particularly, determinants help us understand matrices in terms of volume and coordinate changes, providing awareness about the invertibility of a matrix. If a matrix's determinant is zero, it implies lack of an inverse, and thus, the system of equations it represents has no unique solution.
By mastering these fundamental concepts, students can advance to more complex topics, such as eigenvalues and eigenvectors, which have profound implications in areas like data analysis, physics simulations, and more.