Problem 7 - Precalculus Mathematics for Calculus

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 type of matrix

The given matrix is a 2x2 matrix. It has two rows and two columns. For a 2x2 matrix, the determinant can be calculated directly using a specific formula.

Step 2: Recall the determinant formula for a 2x2 matrix

The determinant of a 2x2 matrix \[\begin{bmatrix} a & b \ c & d\end{bmatrix}\]is given by the formula: \[\text{det}(A) = ad - bc\]

Step 3: Substitute the values into the formula

In the given matrix, \(a = 4\), \(b = 5\), \(c = 0\), and \(d = -1\). Replace these in the formula: \[\text{det}(A) = (4)(-1) - (5)(0)\]

Step 4: Perform the calculation

Calculate the expression: \[\text{det}(A) = 4(-1) - 5(0) = -4 - 0 = -4\]

Step 5: Interpret the result

The determinant of the matrix is \(-4\), which is a non-zero number. Since it is non-zero, the matrix is invertible, and thus its determinant exists.

Key Concepts

2x2 matrix

A 2x2 matrix is a simple yet fundamental concept in linear algebra. It consists of two rows and two columns, forming a square matrix. Notably, 2x2 matrices are the building blocks for understanding more complex matrices and concepts in mathematics.
To help visualize a 2x2 matrix, consider it as a grid. The grid holds four values arranged as follows:

• The first row contains two elements: typically labeled as 'a' and 'b.'
• The second row also contains two elements: commonly referred to as 'c' and 'd.'

An easy way to represent the matrix is with brackets, like this: \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]With its straightforward structure, the 2x2 matrix allows for simple calculations like finding determinants and inverses.

matrix inversion

Matrix inversion is a crucial operation in linear algebra. It involves finding a matrix that, when multiplied by the original matrix, yields the identity matrix. The identity matrix for a 2x2 matrix looks like this:\[ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]Inversion is possible only if the matrix has a non-zero determinant. The inverse of a 2x2 matrix \(A\) is given by the formula:

• If \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), then \(A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\)

Here, \(ad-bc\) is the determinant. If the determinant is zero, the matrix doesn't have an inverse. In the example, since the determinant is \(-4\), an inverse exists, making it a vital part of solving systems of equations and other applications. Remember that not all matrices are invertible, and checking the determinant is the first step to be sure.

linear algebra

Linear algebra is a branch of mathematics that focuses on vectors, matrices, and linear transformations. It's fundamental for understanding mathematical and computational concepts used in various fields.
A key aspect of linear algebra is the study and manipulation of matrices, like the 2x2 matrix discussed earlier. These matrices can represent linear equations, transformations, and more. Different operations used in linear algebra include:

• Matrix addition and subtraction: Combining matrices by adding or subtracting their corresponding elements.
• Scalar multiplication: Multiplying each element of a matrix by a scalar (a single number).
• Matrix multiplication: More complex than addition, this combines two matrices to produce a new matrix.
• Finding the determinant: A scalar value that can determine many properties of a matrix, like whether it is invertible.

Through these operations, linear algebra allows for solving systems of linear equations, analyzing vector spaces, and much more. It's an essential tool for science, engineering, computer graphics, and many other domains, making it incredibly relevant in both academics and industry.