Problem 14 - Algebra 2

Question

Find the value of each determinant. $$ \left|\begin{array}{rr}{-9} & {0} \\ {-12} & {-7}\end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 63.

Step 1: Understand the 2x2 Determinant Formula

For a 2x2 matrix given by \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ad - bc \). This formula is fundamental for determining the value of 2x2 matrices.

Step 2: Identify Matrix Components

From the given matrix \( \begin{bmatrix} -9 & 0 \ -12 & -7 \end{bmatrix} \), identify the components: \( a = -9 \), \( b = 0 \), \( c = -12 \), and \( d = -7 \).

Step 3: Substitute into the Determinant Formula

Apply the components to the formula: \( ad - bc \). Substitute the values to get \( (-9)(-7) - (0)(-12) \).

Step 4: Perform the Calculations

Calculate each part: \((-9)(-7) = 63\) and \((0)(-12) = 0\). Now, subtract the two results: \(63 - 0 = 63\).

Step 5: State the Determinant Result

The determinant of the given matrix is \( 63 \).

Key Concepts

Matrix Algebra

Matrix algebra is a branch of mathematics dedicated to the study and manipulation of matrices. A matrix is essentially an array of numbers arranged in rows and columns. In matrix algebra, we perform operations such as addition, subtraction, and multiplication. Working with matrices helps us solve systems of equations, transform geometric data, and handle complex computations.

• Addition & Subtraction: Matrices can be added or subtracted only when they have the same dimensions.
• Multiplication: Matrices can usually be multiplied if the number of columns in the first matrix matches the number of rows in the second.
• Determinants, as discussed later, are used to find properties such as invertibility.

Understanding matrix algebra is critical because it lays the groundwork for more advanced concepts in linear algebra.

2x2 Matrix

A 2x2 matrix is the simplest form of a square matrix, consisting of two rows and two columns. It is often represented as:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]
Each of the four elements is a number representing data. Square matrices are significant because they possess unique properties, such as the ability to calculate determinants and inverses.
2x2 matrices appear frequently in linear transformations and basic physics problems. They provide a straightforward way to learn about matrix operations and their practical applications.

Determinant Formula

The determinant of a matrix provides valuable information about the matrix's properties. For a 2x2 matrix, the determinant is particularly easy to compute using the formula:\[\text{Determinant} = ad - bc\]
This formula involves the elements of the matrix as follows, where:

• \(a\) and \(d\) are the elements on the main diagonal of the matrix.
• \(b\) and \(c\) are the elements on the off-diagonal.

Hence, the determinant is calculated by multiplying the diagonals and subtracting their products.
A determinant of zero indicates that the matrix is singular, meaning it does not have an inverse. For a non-zero determinant, the matrix has full rank, confirming that it can be inverted. The concept of a determinant plays a crucial role in solutions of linear systems and in understanding matrix invertibility.