Problem 7 - Algebra and Trigonometry

Question

Find the determinant of the matrix, if it exists. $$ \left[\begin{array}{rr}{4} & {5} \\ {0} & {-1}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the matrix is -4.

Step 1: Identify the Matrix

The given matrix is a 2x2 matrix: \( \begin{bmatrix} 4 & 5 \ 0 & -1 \end{bmatrix} \). This matrix is of the form \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \).

Step 2: Apply the Determinant Formula

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula \( ad - bc \).

Step 3: Substitute Values into the Formula

Substitute the given values from the matrix into the formula: \( a = 4 \), \( b = 5 \), \( c = 0 \), and \( d = -1 \). Hence, the determinant is calculated as: \( 4(-1) - 5(0) \).

Step 4: Solve the Expression

Calculate the expression: \( 4(-1) = -4 \) and \( 5(0) = 0 \). Therefore, the determinant is \(-4 - 0 = -4\).

Key Concepts

2x2 Matrix

A **2x2 matrix** is a rectangular array of numbers with two rows and two columns. It is one of the simplest forms of matrices used in matrix algebra. Each number within the matrix is called an element. In the example provided from the exercise, the matrix is:

• Row 1: The elements are 4 and 5.
• Row 2: The elements are 0 and -1.

Matrices are often used to organize data or complex calculations, such as transformations in geometry and solutions to systems of linear equations. They can be identified easily by their dimensions, which in this case are 2x2. Understanding the structure of matrices is foundational in linear algebra, helping students approach more complex manipulations like matrix multiplication or finding determinants.

Determinant Formula

The **determinant formula** is a critical concept in calculating the deterministic value for a matrix. For a 2x2 matrix given by:

• \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]

The determinant is computed using the formula: \( ad - bc \). This means you multiply the top left element \(a\) with the bottom right element \(d\), and then subtract the product of the top right \(b\) and bottom left \(c\) elements.
In simple terms, the determinant of a matrix can provide information about the matrix, such as whether it is invertible or if it transforms space by flipping orientations. Computing determinants correctly is crucial in many applications of linear algebra.

Matrix Algebra

**Matrix algebra** involves various operations you can perform on matrices like addition, subtraction, multiplication, and finding determinants or inverses. It's a powerful tool used throughout mathematics and its applications, including engineering, physics, and computer science.
Understanding the determinant of a matrix like in this exercise helps solve more complex problems. In our case, finding the determinant tells us how the matrix will scale areas in the plane or if it has an inverse.

• If the determinant is zero, the matrix does not have an inverse.
• A non-zero determinant indicates an invertible matrix.

Matrix algebra allows us to explore rotations, scalings, translations, and other transformations in multidimensional space. It enables the conversion of complex systems into manageable computations in real-world problems.